diff --git "a/19A0T4oBgHgl3EQfMv_l/content/tmp_files/load_file.txt" "b/19A0T4oBgHgl3EQfMv_l/content/tmp_files/load_file.txt"
new file mode 100644--- /dev/null
+++ "b/19A0T4oBgHgl3EQfMv_l/content/tmp_files/load_file.txt"
@@ -0,0 +1,1145 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf,len=1144
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='02138v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='CO]  5 Jan 2023  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' EXCLUDING A FOREST IN (THETA, PRISM)-FREE GRAPHS  TARA ABRISHAMI∗†, BOGDAN ALECU∗∗¶, MARIA CHUDNOVSKY∗∐, SEPEHR HAJEBI §,  AND SOPHIE SPIRKL§∥  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large  treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if  H is a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Introduction  All graphs in this paper are finite and simple unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G, H be graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We say that G contains H if G has an induced subgraph isomorphic to H, and we say G is  H-free if G does not contain H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a family H of graphs we say G is H-free if G is H-free  for every H ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A class of graphs is hereditary if it is closed under isomorphism and taking  induced subgraphs, or equivalently, if it is the class of all H-free graphs for some other family  H of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For a graph G = (V (G), E(G)), a tree decomposition (T, χ) of G consists of a tree T and a  map χ : V (T) → 2V (G) with the following properties:  (i) For every v ∈ V (G), there exists t ∈ V (T) such that v ∈ χ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (ii) For every v1v2 ∈ E(G), there exists t ∈ V (T) such that v1, v2 ∈ χ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (iii) For every v ∈ V (G), the subgraph of T induced by {t ∈ V (T) | v ∈ χ(t)} is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For each t ∈ V (T), we refer to χ(t) as a bag of (T, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The width of a tree decomposition  (T, χ), denoted by width(T, χ), is maxt∈V (T) |χ(t)| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The treewidth of G, denoted by tw(G),  is the minimum width of a tree decomposition of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Treewidth was first popularized by Robertson and Seymour in their graph minors project,  and has attracted a great deal of interest over the past three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Particularly, graphs of  bounded treewidth have been shown to be well-behaved from structural [19] and algorithmic [6]  viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This motivates investigating the structure of graphs with large treewidth, and especially,  the substructures emerging in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The canonical result in this realm is the Grid Theorem  of Robertson and Seymour [19], the following, which describes the unavoidable subgraphs of  graphs with large treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a positive integer t, the (t × t)-wall, denoted by Wt×t, is a  planar graph with maximum degree three and treewidth t (see Figure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' a formal definition can  be found in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ∗Princeton University, Princeton, NJ, USA  ∗∗School of Computing, University of Leeds, Leeds, UK  §Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario,  Canada  † Supported by NSF-EPSRC Grant DMS-2120644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ∐ Supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-0083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ¶ Supported by DMS-EPSRC Grant EP/V002813/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ∥ We acknowledge the support of the Natural Sciences and Engineering Research Council of  Canada (NSERC), [funding reference number RGPIN-2020-03912].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Cette recherche a été financée  par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de  référence RGPIN-2020-03912].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This project was funded in part by the Government of Ontario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  1  2  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' W5×5  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 (Robertson and Seymour [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every integer t ≥ 1 there exists w = w(t) ≥ 1  such that every graph of treewidth more than w contains a subdivision of Wt×t as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 can also be reformulated into a full characterization of unavoidable minors in  graphs of large treewidth, that every graph of sufficiently large treewidth contains any given  planar graph as a minor (and no non-planar graph has this property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In contrast, unavoidable  induced subgraphs of graphs with large treewidth are far from completely understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' There  are some natural candidates though, which we refer to as the “basic obstructions”: complete  graphs and complete bipartite graphs, subdivided walls mentioned above, and line graphs of  subdivided walls, where the line graph L(F) of a graph F is the graph with vertex set E(F),  such that two vertices of L(F) are adjacent if and only if the corresponding edges of F share an  end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that the complete graph Kt+1, the complete bipartite graph Kt,t, and the line graph  of every subdivision of Wt×t all have treewidth t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a positive integer t, let us say a graph H  is a t-basic obstruction if H is one of the following graphs: Kt, Kt,t, a subdivision of Wt×t, or  the line graph of a subdivision of Wt×t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We say a graph G is t-clean if G does not contain a  t-basic obstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The basic obstructions do not form a comprehensive list of induced subgraph obstructions  for bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Equivalently, there are t-clean graphs of arbitrarily large treewidth for  small values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A well-known hereditary class of graphs evidencing this fact is the class of  even-hole-free graphs, where a hole is an induced cycle on at least four vertices, the length of  a hole is its number of edges and an even hole is a hole with even length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In fact, for every  positive integer t ≥ 1, one may observe that an even-hole-free graph is t-clean if and only if it is  Kt-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It is therefore tempting to ask whether even-hole-free graphs excluding a fixed complete  graph have bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Sintiari and Trotignon [20] answered this with a vehement no,  providing a construction of (even-hole, K4)-free graphs with arbitrarily large treewidth, hence  proving that there are t-clean (even-hole-free) graphs of arbitrarily large treewidth for every  fixed t ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In addition, graphs from this construction are rather sparse, in the sense that they  exclude short holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 (Sintiari and Trotignon [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers w, l ≥ 1, there exists an (even-hole,  K4)-free graph Gw,l of treewidth more than w and with no hole of length at most l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Note that t-clean graphs for t ≤ 2 have empty vertex set or edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But one might still  hope for 3-clean graphs to have bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This is in fact supported by a result from  [7] asserting that 3-clean even-hole-free graphs have treewidth at most five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' However, another  construction by Sintiari and Trotignon [20] shows that being 3-clean fails to guarantee bounded  treewidth in the more general class of theta-free graphs (see the next section for the definition of  a theta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' one may check that the every t-basic obstruction for t ≥ 3 contains either a theta or a  triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Indeed, the treewidth of theta-free graphs remains unbounded even when forbidding  short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 (Sintiari and Trotignon [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers w, g ≥ 1, there exists a theta-free  graph Gw,g of treewidth more than w and girth more than g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  A natural question to ask then is what further conditions must be imposed to force bounded  treewidth in even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For instance, graphs from both Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 have  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  3  vertices of arbitrarily large degree, and so it was conjectured in [1] that (theta, triangle)-  free graphs of bounded maximum degree have bounded treewidth and even-hole-free graphs  of bounded maximum degree have bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' These were proved in [3] and [4], respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In the same paper [1], a stronger conjecture was made, asserting that basic obstructions  are in fact the only obstructions to bounded treewidth in graphs of bounded maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This was later proved in [16], which closed the line of inquiry into graph classes of bounded  maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4 (Korhonen [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers t, δ ≥ 1, there exists w = w(t, δ) such that every  t-clean graph of maximum degree at most δ has treewidth at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Despite its generality, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4 is surprisingly short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' However, the case of  proper hereditary classes containing graphs of unbounded maximum degree seems to be much  harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For graph classes G and H, let us say H modulates G if for every positive integer t,  there exists a positive integer w(t) (depending on G and H) such that every t-clean H-free graph  in G has treewidth at most w(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' An induced-subgraph analogue to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 is therefore  equivalent to a full characterization of graph classes H which modulate the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This remains out of reach, but the special case where |H| = 1 turns out to be more approachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For a graph H and a graph class G, let us say H modulates G if {H} modulates G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Building  on a method from [17], recently we characterized all graphs H which modulate the class of all  graphs:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then  H modulates the class of all graphs if and only if H is a subdivided star forest, that is, a forest  in which every component has at most one vertex of degree more than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In general, for a hereditary class G containing t-clean graphs of arbitrarily large treewidth for  small t, one may ask for a characterization of graphs H modulating G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Given Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2, a  natural class G to consider is the class of even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 shows  that a graph H modulates even-hole-free graphs only if H is a chordal graph (that is, a graph  with no hole) of clique number at most three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' As far as we know, the converse may also be  true, that every chordal graph of clique number at most three modulates even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In fact, in this paper we narrow the gap, showing that every chordal graph of clique number at  most two, that is, every forest, modulates the class of even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every forest H and every integer t ≥ 1, every even-hole-free graph of suffi-  ciently large treewidth contains either H or a clique of cardinality t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This aligns with the observation [21] that every forest is contained in some graph Gw,l from  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' As mentioned above, one way to improve on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6 is to push H towards  being an arbitrary chordal graph of clique number three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Another way to strengthen Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6  is to find a superclass G of even-hole-free graphs for which forests are the only graphs modulating  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' While the former remains open, we provide an appealing answer to the latter: our main result  shows that forests are exactly the graphs which modulate the class of (theta, prism)-free graphs  (see the next section for the definition of a prism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' again one may check that in (theta, prism)-free  graphs, being t-clean is equivalent to being Kt-free for every positive integer t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then H modulates (theta, prism)-free graphs if and only if  H is a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In other words, given a graph H, for every integer t ≥ 1, every (theta, prism)-free  graph of sufficiently large treewidth contains either H or a clique of cardinality t, if and only if  H is a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let C be the class of all (theta, prism)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It is easily seen that C contains all even-  hole-free graphs, and so Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='7 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that the “only if” direction  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='7 follows immediately from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 as prisms contain triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since every  forest is an induced subgraph of a tree, in order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='7, it suffices to prove  4  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8 below, which we do in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a positive integer t and a tree F, we denote  by Ct the class of all graphs in C with no clique of cardinality t (that is, t-clean graph in C), and  by Ct(F) the class of all F-free graphs in Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1  such that every graph in Ct(F) has treewidth at most τ(F, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We conclude this introduction by sketching our proofs (the terms we use here are defined in  later sections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8 begins with a two-step preparation which culminates  in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2, a result we will also use in subsequent papers in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' As the  first step, inspired by a result from [9], we show that for every graph G ∈ C which contains a  pyramid with certain conditions on the apex and its neighbors, G admits a construction which we  call a “(T, a)-strip-structure,” where a is the apex of the pyramid and T is an optimally chosen  tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Roughly speaking, we show that G\\{a} can be partitioned into two induced subgraphs H  and J where H is more or less similar to the line graph of the tree T and every vertex in J with a  neighbor in H attaches at a pyramid lurking in H in a restricted way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' we call the latter vertices  “jewels”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The proof of this theorem occupies Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The second step is to employ the  previous result to show that if G ∈ Ct admits a (C, a)-strip-structure where C is a caterpillar,  then every vertex in G \\ NG[a] can be separated from a by removing a few vertices (our proof  works more generally when C is any tree of bounded maximum degree, but the caterpillar case  suffices for our application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We prove this in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The central difficulty in the proof is to  deal with the jewels separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This is surmounted in Section 5 where we prove several results  concerning the properties of jewels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Most notably, we show that jewels only attach at “local  areas of the line-graph-like part” of G, and that only a few jewels attach at each local area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This  concludes the preparation for proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Next, we embark on the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We assume that G ∈ Ct has large treewidth,  which together with results from Section 2 implies that G contains two vertices x, y joined by  many pairwise internally disjoint induced paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Now we analyze the structure of  the graph G[P1 ∪ · · · ∪ Pm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It turns out that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' if m is large enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' then either  there are many paths among Pi’s whose union H admits a (C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' x)-strip-structure for some  caterpillar C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' or  for some large value of d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' G[P1 ∪ · · · ∪ Pm] contains a tree S isomorphic to the complete  bipartite graph K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' such that x is the vertex of degree d in S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' and for every leaf l of S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  there are many pairwise internally disjoint induced paths between l and y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' such that in  addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' paths corresponding to distinct leaves of S are also pairwise internally disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The former case implies that y can be separated from x by removing few vertices, which using  a result from Section 6, yields a contradiction with Menger’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The latter case is the first  step towards building the large tree in G as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We now iterate the argument we just  described, applying it to each leaf l of S and y, obtaining larger and larger trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The process is  stopped once we reach a sufficiently large tree as a subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This, combined with the fact  that G ∈ Ct and a result of Kierstead and Penrice [15], yields the desired tree F as an induced  subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Section 2 covers preliminary definitions as well as some  results from the literature used in our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Section 3 investigates the behavior of pyramids in  graphs from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Section 4 is devoted to defining strip-structures and jewels, and showing how they  arise from pyramids in graphs in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Section 5 takes a closer look at jewels for the strip-structures  obtained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In Section 6 we show that admitting certain strip-structures weakens the  connectivity of most vertices to the apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Finally, in Section 7, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Preliminaries and results from the literature  Let G = (V (G), E(G)) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a set X ⊆ V (G) we denote by G[X] the subgraph of  G induced by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For X ⊆ V (G)∪E(G), G\\X denotes the subgraph of G obtained by removing  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  5  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that if X ⊆ V (G), then G \\ X denotes the subgraph of G induced by V (G) \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In  this paper, we use induced subgraphs and their vertex sets interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let x ∈ G and d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We denote by N d  G(x) the set of all vertices in G at  distance d from some x, and by N d  G[x] the set of all vertices in G at distance at most d from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We write NG(x) for N 1  G(x) and NG[x] for N 1  G[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For an induced subgraph H of G, we define  NH(x) = NG(x) ∩ H, NH[x] = NG[x] ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, for X ⊆ G, we denote by NG(X) the set of all  vertices in G \\ X with at least one neighbor in X, and define NG[X] = NG(X) ∪ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let X, Y ⊆ G be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We say X is complete to Y if all edges with an end in X and an  end in Y are present in G, and X is anticomplete to Y if there are no edges between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  A path in G is an induced subgraph of G that is a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  If P is a path in G, we write  P = p1- · · · -pk to mean that V (P) = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , pk} and pi is adjacent to pj if and only if |i−j| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We call the vertices p1 and pk the ends of P, and say that P is from p1 to pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The interior of  P, denoted by P ∗, is the set P \\ {p1, pk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The length of a path is its number of edges (so a path  of length at most one has empty interior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Similarly, if C is a cycle, we write C = c1- · · · -ck-c1  to mean that V (C) = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , ck} and ci is adjacent to cj if and only if |i − j| ∈ {1, k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The  length of a cycle is its number edges (or equivalently, vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=')  A theta is a graph Θ consisting of two non-adjacent vertices a, b, called the ends of Θ, and  three pairwise internally disjoint paths P1, P2, P3 from a to b of length at least two, called the  paths of Θ, such that P ∗  1 , P ∗  2 , P ∗  3 are pairwise anticomplete to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a graph G, by a  theta in G we mean an induced subgraph of G which is a theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  A prism is a graph Π consisting of two disjoint triangles {a1, a2, a3}, {b1, b2, b3} called the  triangles of Π, and three pairwise disjoint paths P1, P2, P3 called the paths of Π, where Pi has  ends ai, bi for each i ∈ {1, 2, 3}, and for distinct i, j ∈ {1, 2, 3}, aiaj and bibj are the only edges  between Pi and Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a graph G, by a prism in G we mean an induced subgraph of G which  is a prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  A pyramid is a graph Σ consisting of a vertex a, a triangle {b1, b2, b3} and three paths P1, P2, P3  of length at least one with Pi from a to bi for each i ∈ {1, 2, 3} and otherwise pairwise disjoint,  such that for distinct i, j ∈ {1, 2, 3}, bibj is the only edge between Pi \\ {a} and Pj \\ {a}, and  at most one of P1, P2, P3 has length exactly one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We say that a is the apex of the pyramid and  b1b2b3 is the base of the pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The pyramid Σ is said to be long if Pi has length more than  one for every i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a graph G, by a pyramid in G we mean an induced subgraph of  G which is a pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Theta, pyramid and prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The dotted lines represent paths of  length at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let us now mention a few results from the literature which we will use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let  G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By a separation in G we mean a triple (L, M, R) of pairwise disjoint subsets of  vertices in G with L ∪ M ∪ R = G, such that neither L nor R is empty and L is anticomplete  to R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let x, y ∈ G be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We say a set M ⊆ G \\ {x, y} separates x and y if there  exists a separation (L, M, R) in G with x ∈ L and y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, for disjoint sets X, Y ⊆ G, we  say a set M ⊆ G \\ (X ∪ Y ) separates X and Y if there exists a separation (L, M, R) in G with  X ⊆ L and Y ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' If X = {x}, we say that M separates x and Y to mean M separates X and  Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Recall the following well-known theorem of Menger [18]:  6  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 (Menger [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let k ≥ 1 be an integer, let G be a graph and let x, y ∈ G be  distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then either there exists a set M ⊆ G \\ {x, y} with |M| < k such that  M separates x and y, or there are k pairwise internally disjoint paths in G from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let k be a positive integer and let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A strong k-block in G is a set B of at least k  vertices in G such that for every 2-subset {x, y} of B, there exists a collection P{x,y} of at least  k distinct and pairwise internally disjoint paths in G from x to y, where for every two distinct  2-subsets {x, y}, {x′, y′} ⊆ B of G, and every choice of paths P ∈ P{x,y} and P ′ ∈ P{x′,y′}, we  have P ∩ P ′ = {x, y} ∩ {x′, y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For a tree T and xy ∈ E(T), we denote by Tx,y the component of T − xy containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G  be a graph and (T, χ) be a tree decomposition for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every S ⊆ T, let χ(S) = �  x∈S χ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By an adhesion of (T, χ) we mean the set χ(x) ∩ χ(y) = χ(Tx,y) ∩ χ(Ty,x) for some xy ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For every x ∈ V (T), by the torso at x, denoted by ˆχ(x), we mean the graph obtained from  the bag χ(x) by, for each y ∈ NT (x), adding an edge between every two non-adjacent vertices  u, v ∈ χ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In [2], we used Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4 and the following result from [13]:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 (Erde and Weißauer [13], see also [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let r be a positive integer, and let G  be a graph containing no subdivision of Kr as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then G admits a tree decomposition  (T, χ) for which the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Every adhesion of (T, χ) has cardinality less than r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For every x ∈ V (T), either ˆχ(x) has fewer than r2 vertices of degree at least 2r4, or ˆχ(x)  has no minor isomorphic to K2r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let k, t ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Then there exists an integer w = w(k, t) ≥ 1 such that every t-clean graph with no strong k-block  has treewidth at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Note that for every t ≥ 3, every subdivision of Wt×t contains a theta and the line graph of  every subdivision of Wt×t contains a prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that for every t ≥ 1, every graph in Ct is  t-clean, and so the following is immediate from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers k, t ≥ 1, there exists an integer β = β(k, t) such that every  graph in Ct with no strong k-block has treewidth at most β(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  A vertex v in a graph G is said to be a branch vertex if v has degree more than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By  a caterpillar we mean a tree C with maximum degree three such that there is a path P in  C containing all branch vertices of C (our definition of a caterpillar is non-standard for two  reasons: a caterpillar is often allowed to be of arbitrary maximum degree, and the path P from  the definition often contains all vertices of degree more than one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By a subdivided star we mean  a graph isomorphic to a subdivision of the complete bipartite graph K1,δ for some δ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In  other words, a subdivided star is a tree with exactly one branch vertex, which we call its root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For every graph H, a vertex v of H is said to be simplicial if NH(v) is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We denote by  Z(H) the set of all simplicial vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that for every tree T, Z(T) is the set of all  leaves of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' An edge e of a tree T is said to be a leaf-edge of T if e is incident with a leaf of  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that if H is the line graph of a tree T, then Z(H) is the set of all vertices in H  corresponding to the leaf-edges of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The following is proved in [2] based on (and refining) a  result from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every integer h ≥ 1,  there exists an integer µ = µ(h) ≥ 1 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a connected graph  with no clique of cardinality h and let S ⊆ G such that |S| ≥ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then either some path in G  contains h vertices from S, or there is an induced subgraph H of G with |H ∩ S| = h for which  one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  H is either a caterpillar or the line graph of a caterpillar with H ∩ S = Z(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  H is a subdivided star with root r such that Z(H) ⊆ H ∩ S ⊆ Z(H) ∪ {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Jumps and jewels on pyramids with trapped apices  For a graph G, an induced subgraph H of G and a vertex a ∈ H, we say a is trapped in H if  we have N 2  G[a] ⊆ H, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  every vertex in NH(a) = NG(a) has degree two in H (and so in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The goal of this section is, for a graph G ∈ C, H ⊆ G and a pyramid Σ in H, to investigate the  adjacency between Σ and a path in G \\ H, assuming that the apex of Σ is trapped in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This  will be of essential use in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We begin with a few definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a graph and let Σ be a pyramid in G with apex  a, base b1b2b3 and paths P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A set X ⊆ Σ is said to be local (in Σ) if either X ⊆ Pi for  some i ∈ {1, 2, 3} or X ⊆ {b1, b2, b3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let P be a path in G \\ Σ with (not necessarily distinct)  ends p1, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For i ∈ {1, 2, 3}, we say P is a corner path for Σ at bi if  p1 has at least one neighbor in Pi \\ {bi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  p2 is complete to {b1, b2, b3} \\ {bi}, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  except for the edges between {p1, p2} and Σ described in the above two bullets, there is  no edge with an end in P and an end in Σ \\ {bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By a corner path for Σ we mean a corner path for Σ at one of b1, b2 or b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let p ∈ G \\ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then p is said to be narrow for Σ if NΣ(p) is local in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Otherwise, we say  p is wide for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For i ∈ {1, 2, 3}, we say p is a jewel for Σ at bi if p is anticomplete to Pi (in  particular, p is anticomplete to a), and for every j ∈ {1, 2, 3} \\ {i}, we have NPj(p) = NPj[bj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By a jewel for Σ we mean a jewel for Σ at one of b1, b2 or b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that if p is either a corner  path or a jewel for Σ, then p is wide for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The following lemma establishes a converse to this  fact for graphs in C and pyramids with a trapped apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ C be graph, let H ⊆ G and let a ∈ H be trapped in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Σ be a pyramid  in H with apex a, base b1b2b3 and paths P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let p ∈ G \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then p is wide for Σ if and  only if p is either a corner path for Σ or a jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We only need to prove the “only if” direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Assume that p ∈ G \\ H is wide for Σ and  p is not a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since a is trapped in H and p ∈ G \\ H, it follows that Σ is long  and p is anticomplete to NΣ[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' First, we show that:  (1) There exists i ∈ {1, 2, 3} for which p is anticomplete to Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose for a contradiction that p has a neighbor in each of P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since p is wide for Σ  and p is not a corner path for Σ, we may assume without loss of generality that p has a neighbor  in P ∗  1 and a neighbor in P ∗  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2, 3}, traversing Pi from a to bi, let xi be the first  neighbor of p in Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since a is trapped, it follows that x1 ∈ P ∗  1 , x2 ∈ P ∗  2 and x3 ∈ P3\\NΣ[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But  then there is a theta in G with ends a, p and paths a-Pi-xi-p for i ∈ {1, 2, 3}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This proves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By (1) and without loss of generality, we may assume that p is anticomplete to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that  since p is wide for Σ, it follows that for every j ∈ {1, 2}, p has a neighbor in Pj, and there exists  j ∈ {1, 2} for which p has a neighbor in P ∗  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each j ∈ {1, 2}, traversing Pj from a to bj, let xj  and yj be the first and the last neighbor of p in Pj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we have xj ∈ P ∗  j \\NPj(a)  for some j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In fact, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (2) For every j ∈ {1, 2}, we have xj ∈ P ∗  j \\ NPj(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then since p is wide for Σ, we may assume without loss of generality that p has  a neighbor in P ∗  1 and we have x2 = y2 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now there is a theta in G with ends a, b2 and  paths a-P1-x1-p-b2, a-P2-b2 and a-P3-b3-b2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (3) For every j ∈ {1, 2}, NPj(p) is a clique of cardinal ity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we may assume without loss of generality that either x1 = y1 or x1 and y1  are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By (2), for every j ∈ {1, 2}, we have xj ∈ P ∗  j \\NPj(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore,  8  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  if x1 = y1, then there is a theta in G with ends a, x1 and paths a-P1-x1, a-P2-x2-p-x1 and  a-P3-b3-b1-P1-x1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Thus, x1 and y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now  there is a theta in G with ends a, p and paths a-P1-x1-p, a-P2-x2-p and a-P3-b3-b1-P1-y1-p, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The proof is almost concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By (3), for every j ∈ {1, 2}, we have NPj(p) = {xj, yj} and xj  is adjacent to yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' If yj ∈ P ∗  j for some j ∈ {1, 2}, then there is a prism in G with triangles xjyjp  and b1b2b3 and paths xj-Pj-a-P3-b3, yj-Pj-bj and p-y3−j-P3−j-b3−j, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Hence, we  have yj = bj for every j ∈ {1, 2}, and so p is a jewel corner for Σ at bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  We can now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ C be a graph, let H ⊆ G and let a ∈ H be trapped in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Σ be a  pyramid in H with apex a, base b1b2b3 and paths P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let P be a path in G \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then  one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  NΣ(P) is local in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  P contains a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  P contains a jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose for a contradiction that there exists a path P in G \\ H for which none of the  outcomes of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We choose such a path P with |P| as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows  that NΣ(P) is not local in Σ, NΣ(X) is local in Σ for every connected set X ⊊ P, P contains  no corner path for Σ and P contains no jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, we have |P| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Since a is trapped in H and P ⊆ G\\H, it follows that Σ is long and P is anticomplete to NΣ[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For every i ∈ {1, 2, 3}, let P ′  i = Pi \\ NPi[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since NΣ(P) is not local and P is minimal subject  to this property, we may assume without loss of generality that  NΣ(p1) ⊆ P ′  1 and p1 has a neighbor in P ′  1 \\ {b1}, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  p2 has a neighbor in P ′  2, and either NΣ(p2) ⊆ P ′  2, or NΣ(p2) ⊆ {b1, b2, b3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  It follows from the choice of P that P ∗ is anticomplete to Σ\\{b1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2}, traversing  Pi from a to bi, let xi and yi be the first and the last neighbor of pi in Pi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' So we  have x1 ∈ P ′  1 \\ {b1}, y1 ∈ P ′  1 and x2, y2 ∈ P ′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In fact, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (4) We have x2 ∈ P ′  2 \\ {b2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we have x2 = y2 = b2, and so b2 ∈ NΣ(p2) ⊆ {b1, b2, b3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that if p2  is adjacent to b3, then P is a corner path for Σ at b1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' So p2 is not adjacent  to b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now there is a theta in G with ends a, b2 and paths a-P1-x1-p1-P-p2-b2, a-P2-b2 and  a-P3-b3-b2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In view of (4) and the choice of P, we conclude that P ∗ is anticomplete to Σ, and for every  i ∈ {1, 2}, we have NΣ(pi) = NP ′  i (pi), xi ∈ P ′  i \\ {bi} and yi ∈ P ′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (5) For every i ∈ {1, 2}, xi and yi are distinct and adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we may assume without loss of generality that either x1 = y1 or x1 and  y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In the former case, there is a theta in G with ends a, x1 and  paths a-P1-x1, a-P2-x2-p2-P-p1-x1 and a-P3-b3-b1-P1-x1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that x1  and y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But then there is a theta in G with ends a, p1 and paths  a-P1-x1-p1, a-P2-x2-p2-P-p1 and a-P3-b3-b1-P1-y1-p1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By (5), for every i ∈ {1, 2}, we have NPi(p) = {xi, yi} and xi is adjacent to yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now there is  a prism in G with triangles p1x1y1 and p2x2y2 and paths P, x1-P1-a-P2-x2 and y1-P1-b1-b2-P2-y2,  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Strip structures with an ornament of jewels  The main result of this section, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2, provides a description of the structure of graphs  in C which have an induced subgraph containing a pyramid with a trapped apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We first set up a framework that allows us to think of a pyramid with apex a as a special case  of a construction similar to the line graph of a tree T, which we call a “(T, a)-strip-structure.”  We start with an induced subgraph W of G that admits an “optimal” (T, a)-strip-structure in  G in a certain sense, and show that the rest of the graph fits into the same construction, except  for vertices which are jewels for certain canonically positioned pyramids in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  First, we need to properly define a strip-structure (this is similar to [8], [9] and [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A  tree T is said to be smooth if T has at least three vertices and every vertex of T is either  a branch vertex or a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let G be a graph, let a ∈ G, let T be a smooth tree, and let  η : V (T) ∪ E(T) ∪ (E(T) × V (T)) → 2G\\{a} be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every S ⊆ V (T), we define  η(S) = �  v∈S,e∈E(T[S])(η(v) ∪ η(e)) and η+(S) = η(S) ∪ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every vertex v ∈ V (T), we  define Bη(v) to be the union of all sets η(e, v) taken over all edges e ∈ E(T) incident with v (we  often omit the subscript η unless there is ambiguity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The function η is said to be a (T, a)-strip-structure in G if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S1) For all distinct o, o′ ∈ V (T) ∪ E(T), we have η(o) ∩ η(o′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S2) If l ∈ V (T) is a leaf of T, then η(l) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S3) For all e ∈ E(T) and v ∈ V (T), we have η(e, v) ⊆ η(e) and η(e, v) ̸= ∅ if and only if e is  incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S4) For all distinct edges e, f ∈ E(T) and every vertex v ∈ V (T), η(e, v) is complete to η(f, v),  and there are no other edges between η(e) and η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In particular, if e and f share no end,  the η(e) is anticomplete to η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S5) For every e ∈ E(T) with ends u, v, define η◦(e) = η(e) \\ (η(e, u) ∪ η(e, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then for every  vertex x ∈ η(e), there is a path in η(e) from x to a vertex in η(e, u) with interior contained  in η◦(e), and there is a path in η(e) from x to a vertex in η(e, v) with interior contained in  η◦(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S6) For all v ∈ V (T) and e ∈ E(T), η(v) is anticomplete to η(e) \\ η(e, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In other words, we  have Nη(T)(η(v)) ⊆ Bη(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S7) For every v ∈ V (T) and every connected component D of η(v), we have NBη(v)(D) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (S8) For every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with l,  a is complete to η(e, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, a has no other neighbors in η(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let S ⊆ η(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We say that S is local in η if S ⊆ η(e) for some e ∈ E(T) or S ⊆ Bη(v) ∪ η(v)  for some v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The following lemma shows that every non-local subset contains a 2-subset  (that is, a subset of cardinality two) which is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a graph and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth tree and η be a (T, a)-strip-  structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Assume also that C ⊆ η(T) is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then there is a 2-subset of C  which is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' First, suppose there exists a vertex x ∈ C ∩ η◦(e) for some e ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since C is not  local, there exists y ∈ C \\ η(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Now {x, y} is a 2-subset of C which is not local in η, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Therefore, we may assume that C ⊆ �  v∈V (T)(B(v) ∪ η(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since the empty set is local in η,  we have C ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' thus, we may pick x ∈ C, v ∈ V (T) and e ∈ E(T) such that x ∈ η(e, v) ∪ η(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  If there exists a vertex y ∈ C \\ (η(e) ∪ B(v) ∪ η(v)), then {x, y} is a 2-subset of C which is not  local in η, and so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Consequently, we may assume that C ⊆ η(e) ∪ B(v) ∪ η(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Since C is not local, there exist x′ ∈ η(e) \\ (B(v) ∪ η(v))) and y′ ∈ (B(v) ∪ η(v)) \\ η(e) such that  {x′, y′} ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Now {x′, y′} is a 2-subset of C which is not local in η, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes  the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  In order to state and prove the main result of this section, we need to define several notions  related to strip-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' From here until the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2, let us fix a graph G,  a vertex a ∈ G, a smooth tree T and a (T, a)-strip-structure η in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  10  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For every edge e ∈ E(T) with ends u, v, by an η(e)-rung, we mean a path P in η(e) ⊆ η(T)  for which either |P| = 1 and P ⊆ η(e, u) ∩ η(e, v), or P has an end in η(e, u) \\ η(e, v) and an  end in η(e, v) \\ η(e, u) and we have P ∗ ⊆ η◦(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Equivalently, a path P in η(e) is an η(e)-rung  if P has an end in η(e, u) and an end in η(e, v) and we have |P ∩ η(e, u)| = |P ∩ η(e, v)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It  follows from (S5) that every vertex in η(e) \\ η◦(e) is contained in an η(e)-rung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In particular,  if either η(e, u) ⊆ η(e, v) or η(e, v) ⊆ η(e, u), then η(e, u) = η(e, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' An η(e)-rung is said to be  long if it is of non-zero length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For every edge e ∈ E(T), let ˜η(e) be the set of vertices in η(e) that are not in any η(e)-rung  (so ˜η(e) ⊆ η◦(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=') We say that η is tame if  η(v) = ∅ for every v ∈ V (T), and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ˜η(e) = ∅ for every e ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In other words, η is tame if and only if every vertex in η(T) is in an η(e)-rung for some e ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For a (T, a)-strip-structure η′ in G, we write η ≤ η′ to mean that for every o ∈ V (T)∪E(T)∪  (E(T)×V (T)), we have η(o) ⊆ η′(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We say a (T, a)-strip-structure η is substantial if for every  e ∈ E(T), there exists a long η(e)-rung in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Equivalently, η is substantial if for every edge  e ∈ E(T) with ends u, v, we have η(e, u) ̸= η(e, v), and so η(e, u) \\ η(e, v), η(e, v) \\ η(e, u) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  One may observe that since T has at least three vertices, if η is substantial and η ≤ η′, then η′  is substantial too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We say η is rich if  a is trapped in η+(T), and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with  l, we have |η(e, l)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  It follows that if there exists a rich (T, a)-strip-structure η in G, then T has exactly |NG(a)|  leaves, and for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident  with l and v ∈ V (T) to be the unique neighbor of l in T, we have η(e, v) ∩ η(e, l) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By a seagull in T we mean a triple (v, e1, e2) where v ∈ V (T) and e1, e2 are two distinct  edges of T incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By a claw in T we mean a 4-tuple (v, e1, e2, e3) where v ∈ V (T) and  e1, e2, e3 are three distinct edges of T incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let (v, e1, e2, e3) be a claw in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By an η-pyramid at (v, e1, e2, e3), we mean a pyramid Σ with  apex a, base b1b2b3 and paths P1, P2, P3, satisfying the following for each i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  bi ∈ η(ei, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  There exists a leaf li of T with the following properties:  (1) li belongs to the component of T \\ {ei} not containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (2) Let Λi be the unique path in T from v to li (so ei ∈ E(Λi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then Pi = Γi ∪ {a},  where Γi is a path in �  e∈E(Λi) η(e) such that Ri = Γi ∩ η(ei) is a long η(ei)-rung  and Γi ∩ η(e) is a η(e)-rung for each e ∈ E(Λi) \\ {ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In particular, assuming ui to be the ends of ei distinct from v and ci to be the unique vertex in  NRi(bi) = NPi(bi) for each i ∈ {1, 2, 3}, we have bi ∈ η(ei, v) \\ η(ei, ui) and ci ∈ η(ei) \\ η(ei, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For a branch vertex v ∈ V (T), by an η-pyramid at v we mean an η-pyramid at (v, e1, e2, e3) for  some claw (v, e1, e2, e3) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, by an η-pyramid we mean an η-pyramid at v for some branch  vertex v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that every η-pyramid is a long pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, if η is substantial,  then for every claw (v, e1, e2, e3) in T there is a η-pyramid at (v, e1, e2, e3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let (v, e1, e2) be a seagull in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A vertex p ∈ G\\η+(T) is said to be a jewel for η at (v, e1, e2)  if for some edge e3 ∈ E(T)\\{e1, e2} incident with v, there exists an η-pyramid Σ at (v, e1, e2, e3)  with base b1b2b3 where bi ∈ η(ei, v) for each i ∈ {1, 2, 3}, such that p is a jewel for Σ at b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In  particular, for each i ∈ {1, 2}, p is adjacent to bi and the unique vertex ci in NPi(bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore,  since Σ is an η-pyramid at (v, e1, e2, e3), assuming ui to be the end of ei distinct from v, it  follows that p has a neighbor bi ∈ η(ei, v) \\ η(ei, ui) and a neighbor ci ∈ η(ei) \\ η(ei, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For a vertex v ∈ V (T), by a jewel for η at v we mean a jewel for η at (v, e1, e2) for some  seagull (v, e1, e2) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, by a jewel for η we mean a jewel for η at v for some branch vertex  v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We denote by Jη the set of all jewels for η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that Jη ⊆ G \\ η+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  11  We are now in a position to prove the main result of this section:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ C, let a ∈ G and let T be a smooth tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose that there exists  a tame, substantial and rich (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then there is a substantial and rich  (T, a)-strip-structure ζ in G for which G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let η be a tame, substantial and rich (T, a)-strip-structure in G such that η(T) is maximal  with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let M = G \\ (η+(T) ∪ Jη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (6)  Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and  x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible  subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then there exists {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) such that  xj1 ∈ B(vj1) \\ η(f) and xj2 ∈ (B(vj2) ∪ η(f)) \\ B(vj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2}, let ei ∈ E(T) such that xi ∈ η(ei) (hence e1 ̸= e2) and  si be an end of ei such that there exists a path Λ0 (possibly of length zero) from s1 to s2 in  T \\ {e1, e2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We claim that there is a vertex v ∈ Λ0 such that B(v) ∩ {x1, x2} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose first  that s1 ̸= s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' let v1 be unique neighbor of s1 in Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we have x1 /∈ B(v1) and x2 /∈ B(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Also, since f = s1v1 does not satisfy (6), we have either x1 /∈ B(s1) or x2 /∈ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But then  either v = s1 or v = v1 satisfies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Thus, we may assume that v = s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note  that since neither e1 nor e2 satisfies (6), we have x1 /∈ B(s1) and x2 /∈ B(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In other words,  we have B(v) ∩ {x1, x2} = ∅, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Henceforth, let v be as promised by the  above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2}, let ui be the end of ei distinct from si (hence u1 ̸= u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let  Λ = u1-s1-Λ0-s2-u2 and let u′  1, u′  2 be the neighbors of v in Λ such that Λ traverses u1, u′  1, v, u′  2, u2  in this order (so either of u1 = u′  1 and u2 = u′  2 is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let e′  i = u′  iv for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since  T is smooth, there exists a vertex u′  3 ∈ NT (v) \\ Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' let e′  3 = u′  3v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2, 3}, let Ti be  the component of T \\(NT (v)\\{u′  i}) containing v (so e′  i ∈ E(Ti)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then since B(v)∩{x1, x2} = ∅  and since η is tame and substantial, there exists an η-pyramid Σ at (v, e′  1, e′  2, e′  3) with apex a,  base b1b2b3 and paths P1, P2, P3 such that we have  bi ∈ η(e′  i, v) and Pi \\ {a, bi} ⊆ η(Ti) \\ B(v) for each i ∈ {1, 2, 3}, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  xi ∈ P ∗  i for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In particular, the second bullet above implies that NΣ(P) is not local in Σ and P is not a corner  path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since P ⊆ M, we have P ∩ Jη = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Thus, Σ being an η-pyramid, it follows that  P contains no jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, since η is rich, a is trapped in η+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, applying  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 to G, H = η+(T), a, Σ and P, we deduce that P contains a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  On the other hand, note that by the second bullet above, for every vertex x ∈ Σ \\ {a}, either  {x, x1} or {x, x2} is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' From this, the minimality of |P| and the fact that η is rich,  it follows that P ∗ is anticomplete to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But then P is a corner path for Σ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This  proves (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (7) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and  x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible  subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let f = v1v2 ∈ E(T) and {j1, j2} = {1, 2} be as guaranteed by (6) applied  to P, x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we have Nη(T)(P ∗) ⊆ η(f, vj1) and Nη(T)({p1, p2}) ⊆ η(f)∪B(v1)∪B(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Without loss of generality, we may assume that j1 = 1 and j2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that by  the minimality of |P|, we have Nη(T)(P ∗) ⊆ η(f, v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, one of p1 and p2 has a neighbor  in η(T) \\ (η(f) ∪ B(v1) ∪ B(v2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' say p1 is adjacent to x′  1 ∈ η(T) \\ (η(f) ∪ B(v1) ∪ B(v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For  each i ∈ {1, 2}, let Ti be the component of T \\ {f} containing vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that there exists  j ∈ {1, 2} such that x′  1 ∈ η(Tj) \\ B(vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Assume that |P| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By the minimality of |P|, we  have j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But then P, x′  1 and x2 violate (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We deduce that |P| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now P, x′  1 and x3−j  violate (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  12  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (8) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and  x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as  possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose that there exist {k1, k2} = {1, 2}, f = v1v2 ∈ E(T)  and e1 ∈ E(T) \\ {f} incident with vk1 such that pk1 has a neighbor in η(e1, vk1) and pk2 has a  neighbor in (B(vk2) ∪ η(f)) \\ B(vk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then pk1 is complete to B(vk1) \\ (η(e1, vk1) ∪ η(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Due to symmetry, we may assume that k1 = 1 and k2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let e3 ∈ E(T)\\{e1, f} be incident  with v1 and let b3 ∈ η(e3, v1) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We need to show that p1 is adjacent to b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose  for a contradiction that p1 and b3 are non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let b1 ∈ η(e1, v1) be adjacent to p1 and let  x ∈ (B(v2) ∪ η(f)) \\ B(v1) be adjacent to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T2 be the component of T \\ (NT (v1) \\ {v2})  containing v1 (so f ∈ E(T2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, for each i ∈ {1, 3}, let ui be the end of ei distinct from v1  and let Ti be the component of T \\ (NT (v1) \\ {ui}) containing v1 (so ei ∈ E(Ti)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By (6) and  (7), there exists an edge f ′ = v′  1v′  2 ∈ E(T) such that Nη(T)({p1, p2}) ⊆ η(f ′) ∪ B(v′  1) ∪ B(v′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This, along with the minimality of |P|, implies that p1 is anticomplete to (η(T1)∪η(T3))\\B(v1),  P \\ {p1} is anticomplete to η(T1) ∪ η(T3) and P \\ {p2} is anticomplete to η(T2) \\ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since  p2 has a neighbor x ∈ (B(v2) ∪ η(f)) \\ B(v1) and since η is tame, there exists a path P2 in G  from a to p2 with P ∗  2 ⊆ η(T2) \\ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, for each i ∈ {1, 3}, there exists a path Pi in G  from a to bi with P ∗  i ⊆ η(Ti) \\ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that since η is rich, P anticomplete to NG[a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' in  particular, P1 has length at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now there is a theta in G with ends a and b1 and  paths P1, a-P2-p2-P-p1-b1 and b1-b3-P3-a, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The following is immediate from (8) and the fact that T is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (9) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and  x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible  subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose that there exist {k1, k2} = {1, 2} and f = v1v2 ∈ E(T) such that  xk1 ∈ B(vk1) \\ (η(f)) and xk2 ∈ (B(vk2) ∪ η(f)) \\ B(vk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then pk1 is complete to B(vk1) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We now deduce:  (10) Let D be a component of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then Nη(T)(D) is local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, there exist x1, x2 ∈ Nη(T)(D) such that {x1, x2} is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For each i ∈ {1, 2}, let pi be a neighbor of xi in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since D is connected, there exists a path P  in D ⊆ M from p1 to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In other words, there exists a path P in M with ends p1, p2 along with  x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) such that {x1, x2} is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Now, let P be a path in M  with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2}  is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' So we can  apply (6) to P, x1 and x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' let {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) be as in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We may assume  without loss of generality that j1 = 1 and j2 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' thus, v1 is a branch vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows from  (7) that Nη(T)(P ∗) ⊆ η(f, v1) and Nη(T)({p1, p2}) ⊆ η(f) ∪ B(v1) ∪ B(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By (9) applied to  k1 = 1 and k2 = 2, p1 is complete to B(v1) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, from (9) applied to k1 = 2 and k2 = 1,  it follows that either p2 is complete to B(v2) \\ η(f) and B(v2) \\ η(f) ̸= ∅, or p2 is anticomplete  to B(v2) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that if |P| > 1, then by the minimality of |P|, we have Nη(T)(p1) ⊆ B(v1)  and Nη(T)(p2) ⊆ (B(v2)∪η(f))\\B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let us define η′ : V (T)∪E(T)∪(E(T)×V (T)) ⊆ 2G\\{a}  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let η′(f) = η(f) ∪ P and let η′(f, v1) = η(f, v1) ∪ {p1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let  η′(f, v2) = η(f, v2) ∪ {p2} if p2 is complete to B(v2) \\ η(f) and B(v2) \\ η(f) ̸= ∅, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  η′(f, v2) = η(f, v2) if p2 is anticomplete to B(v2) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let η′ = η elsewhere on V (T) ∪ E(T) ∪ (E(T) × V (T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then since η is tame, substantial and  rich, and p2 is adjacent to x2 ∈ B(v2) ∪ η(f)) \\ B(v1), it is straightforward to check that η′  is also a tame, substantial and rich (T, a)-strip-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But we have η′(T) = η(T) ∪ P, a  contradiction with the maximality of η(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The proof is almost concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let X be the union of all the components D of M such that  D is anticomplete to η+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since η is rich, it follows that for every component D of M \\ X, a  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  13  is anticomplete to X and Nη+(T)(D) = Nη(T)(D) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By (10), for every component D  of M \\ X, Nη(T)(D) is local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let D be the set of all components D of M \\ X for which we  have Nη+(T)(D) ⊆ Bη(v) for some v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Breaking the ties arbitrarily and by the definition  of X, we may write D = �  v∈V (T) Dv, where  for all distinct u, v ∈ V (T), we have Du ∩ Dv = ∅, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  for all v ∈ V (T) and every D ∈ Dv, we have Nη+(T)(D) ⊆ Bη(v) and Nη+(T)(D) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Also, for every e = uv ∈ E(T), let De be the set of all components D of M \\ X for which we  have Nη+(T)(D) ⊆ η(e) and  either Nη(T)(D) ∩ η◦(e) ̸= ∅, or;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Nη(T)(D) ∩ (η(e, u) \\ η(e, v)) ̸= ∅ and Nη(T)(D) ∩ (η(e, v) \\ η(e, v)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  From the definition of X, it follows that every component of M \\ X belongs to exactly one of  the sets {Dv, De : v ∈ V (T), e ∈ E(T)} (note that since η is rich, a is anticomplete to each such  component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let ζ : V (T) ∪ E(T) ∪ (E(T) × V (T)) ⊆ 2G\\{a} be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all v ∈ V (T) and  e ∈ E(T), let  ζ(v) = �  D∈Dv D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ζ(e) = η(e) ∪ (�  D∈De D), and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ζ(e, v) = η(e, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  It is easily seen that ζ satisfies the conditions (S1-S8) from the definition of a (T, a)-strip-  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In particular, since η is rich, ζ satisfies (S2), and from the definitions of X, Dv’s and  De’s, it follows that ζ satisfies (S5) and (S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, we have η ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Now, since η is substantial and rich, since η ≤ ζ and from the definitions of X and ζ, it follows  that ζ is a substantial and rich (T, a)-strip-structure with Jζ = Jη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Moreover, note that we have  ζ+(T) = η(T)+ ∪ (M \\ X), and so G \\ (ζ+(T) ∪ Jζ) = G \\ (ζ+(T) ∪ Jη) = X is anticomplete to  ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Jewels under the loupe  Here we revisit jewels for strip-structures, establishing several results about their proper-  ties in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This will help attune Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 for its application in the proof of  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  First we need to introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth  tree and let ζ be a (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let v ∈ V (T) and let e ∈ E(T) be incident with  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We denote by ζe(v) the set of all components D of ζ(v) for which we have NB(v)(D) ⊆ η(e, v),  or equivalently, Nζ(T)\\ζ(e,v)(D) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let (v, e1, e2) be a seagull in T and let ui be the end of ei distinct from v for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We define  ζ(v, e1, e2) = ζ(e1) ∪ ζ(e2) ∪ ζe1(u1) ∪ ζe2(u2) ∪ ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We denote by Jζ,(v,e1,e2) the set of all jewels for ζ at (v, e1, e2), and for every vertex v ∈ V (T),  Jζ,v stands for the set of all jewels for η at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that Jζ,v = ∅ if v is a leaf of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The first result in this section describes, for a (T, a)-strip-structure in a theta-free graph, the  attachments of jewels at a vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a theta-free graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth tree and let ζ be  a (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let (v, e1, e2) be a seagull in T and let x ∈ Jζ,(v,e1,e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then the  following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We have Nζ+(T)(x) ⊆ ζ(v, e1, e2), and so Nζ+(T)(Jζ,(v,e1,e2)) ⊆ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Consequently,  for every vertex v ∈ V (T), we have Nζ+(T)(Jζ,v) ⊆ ζ(NT [v]), and for every two distinct  vertices v, v′ ∈ V (T), we have Jζ,v ∩ Jζ,v′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  14  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Assume that ζ is rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let i ∈ {1, 2} and let R be a long ζ(ei)-rung, let r be the end of  R in ζ(ei, v) and let r′ be the unique neighbor of r in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then either x is anticomplete  to R or NR(x) = {r, r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that v is a branch vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2}, let ui be the end of ei distinct  from v and let Ti be the component of T \\(NT (v)\\{ui}) containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T ′ be the component  of T \\{u1, u2} containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let x ∈ Jζ,(v,e1,e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since x ∈ Jζ,(v,e1,e2) is a jewel for ζ, there exists  an edge e3 ∈ E(T)\\{e1, e2} incident with v and a ζ-pyramid Σ at (v, e1, e2, e3) with apex a, base  b1b2b3 and paths P1, P2, P3 such that x is a jewel for Σ at b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In particular, for each j ∈ {1, 2, 3},  Pj ∩ ζ(ej) is a long ζ(ej)-rung Rj with bj as its end in ζ(ej, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Also, x is anticomplete to  P3 (and so x is not adjacent to a), and for each j ∈ {1, 2}, assuming cj to be the unique  vertex in NRj(bj) = NPj(bj), x is adjacent to bj ∈ ζ(ej, v) \\ ζ(ej, uj) and cj ∈ ζ(ej) \\ ζ(ej, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Therefore, there exist paths Qi, Si of length more than one in G from a to x for which we have  bi ∈ Q∗  i ⊆ (ζ(T ′) \\ ζ(v)) ∪ (ζ(ei, v) \\ ζ(ei, ui)) and ci ∈ S∗  i ⊆ ζ(Ti) \\ (B(v) ∪ ζ(ui) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  To prove the first assertion of the Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, assume for a contradiction that x has a  neighbor y ∈ ζ+(T) \\ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since x is not adjacent to a, we have y ∈ ζ(T) \\ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  First, assume that y ∈ ζ(T ′)\\ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then by (S5) and (S7) from the definition of a strip-structure,  there exists a path Q′ of length more than one in G from a to x with Q′∗ ⊆ ζ(T ′) \\ ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But  now there is a theta in G with ends a, x and paths a-S1-x, a-S2-x and a-Q′-x, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  It follows that y ∈ ζ(T1 ∪ T2) \\ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In other words, for some i ∈ {1, 2}, we have  y ∈ ζ(Ti) \\ (ζ(ei) ∪ ζei(ui) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' As a result, by (S5) and (S7) from the definition a strip-  structure, and by the definition of ζei(ui), there exists a path S′  i of length more than one in G  from a to x with S′∗  i ⊆ ζ(Ti)\\(ζ(ei)∪ζei(ui)∪ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now assuming i′ ∈ {1, 2} to be distinct  from i, there is a theta in G with ends a, x and paths a-Qi-x, a-S′  i-x and a-Si′-x, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This proves the the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Next we prove the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By symmetry, we may assume that i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Assume that x has a neighbor y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let P ′  1 = (P1 \\ R1) ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Σ′ be the pyramid with  apex a, base rb2b3 and paths P ′  1, P2 and P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Recall that since ζ is rich, a is trapped in ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Also, Σ′ is a pyramid in ζ+(T), x is adjacent to y ∈ P ′  1, x is adjacent to b2, c2 ∈ P2 and x is  anticomplete to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that x is a wide vertex for Σ′ which is not a corner path for Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Now applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 to G, a, H = ζ+(T), Σ′ and p = x, we deduce that x is a jewel for Σ′  at b3, and so NR(x) = NP ′  1(x) = {r, r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  Our next goal is to show that for every rich (T, a)-strip-structure in a graph G ∈ Ct, there  are only a few jewels at each vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let us begin with a lemma, asserting that for a rich  (T, a)-strip-structure ζ in a theta-free graph, each set Bζ(v) is almost a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a theta-free graph and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth tree and ζ be a rich  (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then for every v ∈ V (T), there exists at most one edge f ∈ E(T)  such that η(f, v) is not a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose for a contradiction that there are two distinct edges f1, f2 ∈ E(T) incident with  v, and for each i ∈ {1, 2}, there exist xi, yi ∈ ζ(fi, v) such that xi is not adjacent to yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then  v is not a leaf of T and H = x1-x2-y1-y2-x1 is a hole of length four in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since ζ is rich, a  is anticomplete to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let f1 = u1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let l1 be a leaf of T which belongs to the component of  T \\ {v} containing u1, and let Λ1 be the unique path in T from v to l1 (so f1 ∈ E(Λ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let  Rx1 be an ζ(f)-rung containing x1 and let Ry1 be an ζ(f)-rung containing y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since ζ is rich,  H1 = Rx1 ∪ Rx2 ∪ B(u1) is a connected induced subgraph of G, and so there is a path Q in H1  from x1 to y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that Q has length more than one and Q∗ ⊆ (B(u1) ∪ ζ(f1))\\B(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But  now there is a theta in G with ends x1, y1 and paths Q, x1-x2-y1 and x1-y2-y1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This completes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  Recall the following classical result of Ramsey (see, for instance, [5] for an explicit bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=')  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  15  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 (See [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers a, b ≥ 1, there exists an integer R = R(a, b) ≥ 1 such  that every graph G on at least R(a, b) vertices contains either a clique of cardinality a or a stable  set of cardinality b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We can now prove the second main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all positive integers t, δ, there exists a positive integer j = j(t, δ) with the  following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ Ct be a graph and let a ∈ G and let T be a smooth tree of maximum  degree δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let ζ be a rich (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then for every vertex v ∈ V (T), we have  |Jζ,v| < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let j = j(t, δ) =  �δ  2  �R(t, 3) with R(·, ·) as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Then in order to prove  |Jζ,v| < j, it is enough to show that |Jζ,(v,e1,e2)| < R(t, 3) for every seagull (v, e1, e2) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose for a contradiction that |Jζ,(v,e1,e2)| ≥ R(t, 3) for some seagull (v, e1, e2) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then v is  a branch vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2}, let ui be the end of ei different from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since G ∈ Ct,  it follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 that Jζ,(v,e1,e2) contains a stable set X of cardinality three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For  every x ∈ X, since x is a jewel for ζ at (v, e1, e2), it follows that for every i ∈ {1, 2}, there exists  a long ζ(ei)-rung Rx  i such that Qx  i = Rx  i \\ ζ(ei, v) is a path in ζ(ei) \\ ζ(ei, v) from a neighbor  of x to a vertex in ζ(ei, ui) \\ ζ(ei, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' in particular, Rx  i contains a neighbor of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, for  each i ∈ {1, 2}, we may pick a non-empty set Ri of long ζ(ei)-rungs such that every vertex in X  has a neighbor in at least one rung in Ri, and with Ri minimal with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We  deduce:  (11) There exists i ∈ {1, 2} with |Ri| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then for every i ∈ {1, 2}, there exists a long ζ(ei)-rung Si such that every vertex  in X has a neighbor in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let si be the end of Si in ζ(ei, v) and s′  i be unique neighbor of si in  Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, X is complete to {s′  1, s′  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now X ∪ {s′  1, s′  2} is  a theta in G with ends s′  1, s′  2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By (11) and due to symmetry, we may assume that |R1| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This, together with the  minimality of R1, implies that there exist distinct vertices x, y ∈ X as well as distinct long  ζ(e1)-rungs Rx, Ry ∈ R1 such that x has a neighbor in Rx, y has a neighbor in Ry, x is  anticomplete to Ry, and y anticomplete to Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let rx and ry be the ends of Rx and Ry in  ζ(e1, v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let r′  x be the unique neighbor of rx in Rx and r′  y be the unique neighbor  of ry in Ry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' so we have r′  x, r′  y ∈ ζ(e1) \\ ζ(e1, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, we  have NRx∪Ry(x) = {rx, r′  x} and NRx∪Ry(y) = {ry, r′  y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that rx, r′  x ∈ Rx \\ Ry and  ry, r′  y ∈ Ry \\ Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, rx is anticomplete to Ry \\ {ry}, as otherwise (Ry \\ {ry}) ∪ {rx} contains  a long ζ(e1)-rung R with NR(x) = {rx}, which violates the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Similarly, ry is anticomplete to Rx \\ {rx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Now, let G1 = G[(B(u1)\\ζ(e1, u1))∪((Rx∪Ry)\\{rx, ry})] and let G2 = G[(B(u2)\\ζ(e2, u2))∪  Qx  2 ∪ Qy  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since ζ is rich, the second bullet in the definition of a rich strip-structure implies that  G1 and G2 are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Consequently, there exists a path Q1 in G1 from r′  x to r′  y, and there  exists a path Q2 from x to y with Q∗  2 ⊆ G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, since v is a branch vertex of T, we may choose  an edge e3 ∈ E(T) \\ {e1, e2} incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By the first assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, {x, y} is  anticomplete to ζ(e3, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Q3 be a path from rx to ry with Q∗  3 ⊆ ζ(e3, v) (thus |Q3| ∈ {2, 3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  But now there is a prism with triangles xrxr′  x and yryr′  y and paths Q1, Q2, Q3, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  Our last theorem in this section examines the connectivity within G \\ ζ+(T) for a (T, a)-  strip-structure ζ arising from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We need the following lemma, the proof of which is  similar to that of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G be a theta-free graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth tree and let ζ be a  (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let v, v′ ∈ V (T) be distinct and let P be a path in G \\ ζ+(T) with  16  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ends x, x′ such that x ∈ Jζ,v, x′ ∈ Jζ,v′ and P ∗ is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then v and v′ are  adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, x and x′ are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Λ be the path in T  from v to v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then Λ has length more than one, and so there are two distinct edges f, f ′ ∈ E(Λ)  such that f is incident with v and f ′ is incident with v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let u be the end of f distinct from  v and u′ be the end of f ′ distinct from v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let (v, e1, e2) and (v′, e′  1, e′  2) be two seagulls in G  such that x ∈ Jζ,(v,e1,e2) and x′ ∈ Jζ,(v′,e′  1,e′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ {1, 2}, let ui be the end of ei distinct  from v and let u′  i be the end of e′  i distinct from v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Without loss of generality, we may assume  that u2, u′  2 /∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T2 be the component of T \\ (NT (v) \\ {u2}) containing v and let T ′  2 be the  component of T \\(NT (v′)\\{u′  2}) containing v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T ′ be the component of T \\{u′, u′  2} containing  v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since x is a jewel for ζ at (v, e1, e2), it follows that x is not adjacent to a, and x has a neighbor  c ∈ ζ(e2) \\ ζ(e2, v) ⊆ ζ(T2) \\ (B(v) ∪ ζ(u2) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, there exists a path Q of length  more than one in G from a to x for which we have c ∈ Q∗ ⊆ ζ(T2) \\ (B(v) ∪ ζ(u2) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also,  since x′ is a jewel for ζ at (v′, e′  1, e′  2), it follows that x′ is not adjacent to a, and x′ has a neighbor  b′ ∈ B(v′)\\(ζ(f ′, u′)∪ζ(e′  2, v′)) and a neighbor c′ ∈ ζ(e′  2)\\ζ(e′  2, v′) ⊆ ζ(T ′  2)\\(B(v′)∪ζ(u′  2)∪ζ(v′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Therefore, there exist paths P ′, Q′ of length more than one in G from a to x′ for which we have  b′ ∈ P ′∗ ⊆ (ζ(T ′) \\ ζ(v′)) ∪ (ζ(f ′, v′) \\ ζ(f ′, u′)) and c′ ∈ Q′∗ ⊆ ζ(T ′  2) \\ (B(v′) ∪ ζ(u2) ∪ ζ(v′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  But now there is a theta in G with ends a, x′ and paths a-P ′-x′, a-Q′-x′ and a-Q-x-P-x′, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let t, δ ≥ 1 be integers and let j(t, δ) be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ Ct be a  graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth tree of maximum degree δ and let v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let ζ  be a rich (T, a)-strip-structure in G such that G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let  x ∈ G \\ (ζ+(T) ∪ Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then there exists Sx ⊆ G \\ (ζ+(T) ∪ {x}) such that |Sx| < 2j(t, δ) and Sx  separates x and Jζ \\ ({x} ∪ Sx) in G \\ ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Consequently, Sx separates x and ζ+(T) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, {Jζ,v : v ∈ V (T)} is a partition of Jζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G′ be the graph obtained  from G\\ζ+(T) by contracting the set Jζ,v into a vertex zv for each v ∈ V (T) with Jζ,v ̸= ∅, and  then adding a new vertex z such that NG′(z) = {zv : v ∈ V (T), Jζ,v ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We claim that there  is a set Y ⊆ G′ \\ {x, z} of cardinality at most two which separates x and z in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, there are three pairwise internally disjoint paths in G′ from x to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Thus, there  exist S ⊆ T with |S| = 3 as well as three paths {Pv : v ∈ S} in G \\ ζ+(T) all having x as an end  and otherwise disjoint, such that for each v ∈ S, Pv has an end yv ∈ Jζ,v distinct from x, and  we have P ∗  v ⊆ G\\(ζ+(T) ∪ Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' As a result, for all distinct v, v′ ∈ S, Pv,v′ = yv-Pv-x-Pv′-yv′ is a  path in G\\ζ+(T) from yv ∈ Jζ,v to yv′ ∈ Jζ,v′ such that P ∗  v,v′ ⊆ G\\(ζ+(T)∪Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In particular,  P ∗  v,v′ is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5, S is a clique in T, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let Y be as in the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each y ∈ Y , if y = zv for some v ∈ V (T), then let  Ay = Jζ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Otherwise, let Ay = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Sx = �  y∈Y Ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then Sx ⊆ G\\(ζ+(T)∪{x}) separates  x and Jζ \\({x}∪Sx) in G\\ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4, we have |Sx| < 2j(t, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes  the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Strip structures and connectivity  In this section, we investigate the connectivity implications of the presence of certain (T, a)-  strip-structures in graphs from Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers t, δ ≥ 1, there exists an integer σ = σ(t, δ) ≥ 1 with the following  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ Ct be a graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be a smooth tree of maximum degree δ and  let ζ be a rich (T, a)-strip-structure in G such that G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Then for every vertex x ∈ G \\ NG[a], there exists a set Sx ⊆ G \\ {a, x} with |Sx| < σ such that  S separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let j(t, δ) be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We claim that  σ = σ(t, δ) = 2δ(j(t, δ) + t)  satisfies Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every vertex v ∈ V (T), we define Cv = B(v) if v is a leaf of T and  Cv = ∅ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, for every vertex v ∈ V (T), let Kv be a maximal clique of G contained  in B(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Thus, we have |Kv| < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Moreover, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 along with the assumption that ζ is  rich implies if v is a leaf of T, then we have Kv = B(v) = Cv (and so |Kv| = 1), and if v is a  branch vertex of T, then Kv contains all but possibly one of the sets η(f, v) for f ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For  every S ⊆ T, we define  MS =  �  w∈NT (S)  Jη,w,  NS =  �  w∈NT (S)  Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Also, we write Mv for M{v} and Nv for N{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every v ∈ V (T), let Ov = Mv ∪ Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' The  following is immediate from Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (12) For every v ∈ V (T), we have  Ov ⊆ G \\ (Jζ,v ∪ {a});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  |Ov| < δ(j(t, δ) + t) ≤ σ, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Ov separates a and Jζ,v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Now, for every x ∈ G \\ NG[a], we define Sx as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' First, assume that x ∈ ζ(T) \\ NG[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Then either x ∈ ζ(e) for some edge e = uv ∈ E(T), or x ∈ ζ(v) for some branch vertex v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In the former case, let  Ex = Mu ∪ Mv,  Ix = N{u,v} ∪ Cu ∪ Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In the latter case, let  Ex = Mv ∪ Jζ,v  Ix = Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let Sx = Ex ∪ Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Observe that since x ∈ G \\ NG[a], we have Sx ⊆ G \\ {a, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Also, by  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4, we have |Ex| ≤ 2δj(t, δ) and so |Sx| < 2δ(j(t, δ) + t) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Moreover, from  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 and the fact that ζ is rich, it is easy to check that for every path P in G from a  to x, if P ⊆ ζ+(T), then P contains a vertex from Ix, and otherwise P contains a vertex from  either Ix or Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Next, assume that x ∈ Jζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1, there exists a unique vertex v ∈ V (T) such  that x ∈ Jζ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let Sx = Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then by (12), we have Sx ⊆ G \\ {a, x}, |Sx| < σ and Sx separates  a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Finally, assume that x ∈ G\\(ζ+(T)∪Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then letting Sx to be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6, it follows  from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='6 that Sx ⊆ G \\ {a, x}, |X| < 2j(t, δ) ≤ σ and Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This  completes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  Our application of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 though is confined to the case where T is a caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' More  precisely, for a graph G and a vertex a ∈ G, an induced subgraph H ⊆ G \\ {a} is said to be an  a-seed in G if the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  There exists a caterpillar C such that H is the line graph of a 1-subdivision of C and  NG(a) = Z(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The vertex a is trapped in H ∪ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  It follows that Z(H) is the set of all degree-one vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We now combine Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2  and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 to deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  18  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every integer t ≥ 1, there exists an integer s = s(t) ≥ 1 with the following  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ Ct be a graph and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Assume that there is an a-seed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then for  every vertex x ∈ G \\ NG[a], there exists Sx ⊆ G \\ {a, x} with |Sx| < s such that Sx separates a  and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let σ(·, ·) be as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We show that s = s(t) = σ(t, 3) satisfies Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Pick an a-seed H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let T be the unique smooth caterpillar with |NG(a)| leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then T has  maximum degree three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, one may immediately observe that there is a tame, substantial  and rich (T, a)-strip-structure η in G with η(T) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Now we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 to G, a  and T, deducing that there exists a substantial and rich (T, a)-strip-structure ζ in G such that  G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Hence, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1 applied to G, a, T and ζ,  for every vertex x ∈ G \\ NG[a], there exists Sx ⊆ G \\ {a, x} with |Sx| < s such that Sx separates  a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' From blocks to trees  In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We begin with a result which captures the use of  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For a positive integer n, we write [n] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers ν, t ≥ 1, there exists an integer ψ = ψ(t, ν) ≥ 1 with the  following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ Ct, let a, b ∈ G be distinct and non-adjacent and let {Pi : i ∈ [ψ]}  be a collection of ψ pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For each i ∈ [ψ], let  ai be the neighbor of a in Pi (so ai ̸= b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then there exists I ⊆ [ψ] with |I| = ν for which the  following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  {ai : i ∈ I} ∪ {b} is a stable set in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  For all i, j ∈ I with i < j, ai has a neighbor in P ∗  j \\ {aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let s = s(t) be as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 and let µ = µ(max{2s + 1, t}), where µ(·) is as in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let R(·, ·) be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every integer p ≥ 1, let Rtourn(p) be the  smallest positive integer n such that every tournament on at least n vertices contains a transitive  tournament on p vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' the existence of Rtourn(p) follows easily from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 (in fact,  one may observe that Rtourn(p) ≤ R(p, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let γ = R(Rtourn(ν + 1), µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We prove that  ψ = ψ(t, ν) = R(γ, t)  satisfies Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , Pψ be ψ pairwise internally disjoint paths in G from a to  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since G is Kt-free, it follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 and the definition of ψ that there exists a  stable set N ⊆ {ai : i ∈ [ψ]} in G with |N| = γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' we may assume without loss of generality that  N = {ai : i ∈ [γ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let D be a directed graph with V (D) = N such that for distinct i, j ∈ [γ], there is an arc  from ai to aj in D if and only if xi has a neighbor in P ∗  j \\ {aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that D may contain both  arcs (ai, aj) and (aj, ai), and so the undirected underlying graph of D might not be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let  D− be the simple graph obtained from the undirected underlying graph of D by removing one  of every two parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  (13) D− contains no stable set of cardinality µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Suppose for a contradiction that D− contains a stable set S of cardinality µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We may assume  without loss of generality that S = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , aµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let G1 = G[(�µ  j=1 Pj) \\ {a}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Note that  by the definition of D, for every i ∈ [µ], we have NG1(ai) = NPi(ai) \\ {a}, and in particular  |NG1(ai)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since G1 is connected and Kt-free, and since and |S| = µ = µ(max{2s + 1, t}),  we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='5 to G1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Note that every vertex in S has a unique neighbor in  G1, and so no path in G1 contains max{2s + 1, t} ≥ 3 vertices from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Consequently, there is  an induced subgraph H1 of G1 with |H1 ∩ S| = 2s + 1 for which one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  H1 is either a caterpillar or the line graph of a caterpillar with H1 ∩ S = Z(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  H1 is a subdivided star with root r1 such that Z(H1) ⊆ H1 ∩ S ⊆ Z(H1) ∪ {r1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  19  If H1 is a caterpillar, then G[H1∪{a}] contains a theta with ends a and a′ for every vertex a′ ∈ H1  of degree more than two, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, if the second bullet above holds, then since every  vertex in S is of degree one in G1, we have H1 ∩ S = Z(H1), and so r1 is not adjacent to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But  then G[H1 ∪ {a}] contains a theta with ends x and r1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that H1 is the  line graph of a caterpillar with |H1 ∩ S| = 2s + 1 and H1 ∩ S = Z(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This, together with the  fact that every vertex in H1 ∩ S ⊆ S has a unique neighbor in H1 ⊆ G, implies that H1 contains  the line graph H2 of a 1-subdivision of a caterpillar with |H2 ∩ S| = s and H2 ∩ S = Z(H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let S2 = H2 ∩ S = Z(H2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' then S2 is the set of all vertices of degree one in H2, and we may  assume without loss of generality that S2 = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , as}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G2 = G[H2 ∪(�s  j=1 Pj)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows  that G2 ∈ Ct, NG2(a) = S2 = Z(H2) and a is trapped in H2 ∪ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Therefore, H2 is an a-seed  in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since b ∈ G2 \\ NG2[a], applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 to G2 and a, we deduce that there exists  Sb ⊆ G2 \\{a, b} such that |Sb| < s and Sb separates a and b in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , Ps are s pairwise  internally disjoint paths in G2 from a to b, a contradiction with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This proves (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  By (13), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 and the definition γ, D− contains a clique of cardinality Rtourn(ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  This, along with the definition of Rtourn(·), implies that D contains (as a subdigraph) a transitive  tournament K on ν + 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  We may assume without loss of generality that V (K) =  {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , aν+1} such that for distinct i, j ∈ [ν + 1], (ai, aj) is an arc in K if i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' From the  definition of D, it follows that {a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , aν+1, b} is a stable set in G, and for all i, j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , ν+1}  with i < j, ai has a neighbor in P ∗  j \\{aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Hence, I = {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , ν + 1} satisfies Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  For positive integers d and r, let T r  d denote the rooted tree in which every leaf is at distance  r from the root, the root has degree d, and every vertex that is neither a leaf nor the root has  degree d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We need a result from [15]:  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 (Kierstead and Penrice [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers d, r, s, t ≥ 1, there exists an integer  f = f(d, r, s, t) ≥ 1 such that if G contains T f  f as a subgraph, then G contains one of Ks,s, Kt  and T r  d as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  The following lemma is the penultimate step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For all integers d, r, t ≥ 1, there exists an integer m = m(d, r, t) with the following  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let G ∈ Ct be a graph, let a, b ∈ G be non-adjacent and let {Pi : i ∈ [m]} be a collection  of m pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then G[�m  j=1 Pj] contains a subgraph  J isomorphic to T r  d such that a ∈ J and a has degree d in J (that is, a is the root of J), and we  have b /∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let d, t ≥ 1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let m1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every integer r > 1, let mr = ψ(t, (mr−1 + 1)d)  where ψ(·, ·) is as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We prove by induction on r ≥ 1 that m(d, r, t) = mr satisfies  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , Pmr be mr pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since a  and b are not adjacent, it follows that for each i ∈ [mr], we have P ∗  i ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' let ai be the neighbor  of a in Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' In particular, we have b /∈ {ai : i ∈ [mr]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose first that r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then we have  |{ai : i ∈ [m1]}| = m1 = d, and so G[{ai : i ∈ [mr]} ∪ {a}] contains a (spanning) subgraph  J isomorphic to T 1  d such that a ∈ J and a has degree d in J, and we have b /∈ J, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Therefore, we may assume that r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Since mr = ψ(t, (mr−1 + 1)d), we can apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1  to a, b and {Pi : i ∈ [mr]}, obtaining I ⊆ [mr] with |I| = (mr−1 + 1)d which satisfies the two  outcomes of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Without loss of generality, we may assume that I = [(mr−1 + 1)d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  It follows that {a1, · · · , a(mr−1+1)d, b} is a stable set in G, and for all i, j ∈ [(mr−1 + 1)d] with  i < j, ai has a neighbor in P ∗  j \\ {aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every i ∈ [d], let a′  i = a(i−1)mr−1+i and let  Ai = {(i − 1)mr−1 + i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , (i − 1)mr−1 + i + mr−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  In particular, we have |Ai| = mr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Then for each i ∈ [d] and each j ∈ Ai, a′  i has a neighbor in  P ∗  j \\ {aj}, and so there exists a path Qj in G from a′  i to b with Q∗  j ⊆ P ∗  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Now, for every i ∈ [d],  a′  i and b are non-adjacent, and {Qj : j ∈ Ai} is a collection of mr−1 pairwise internally disjoint  20  INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  paths in G from a′  i to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows from the induction hypothesis that G[�  j∈Ai Qj] contains a  subgraph Ji isomorphic to T r−1  d  such that a′  i ∈ Ji and a′  i has degree d in Ji, and we have b /∈ Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  But now G[(�d  i=1 V (Ji)) ∪ {a}] ⊆ G[�mr  j=1 Pj] contains a (spanning) subgraph J isomorphic to  T r  d such that a ∈ J and a has degree d in J, and we have b /∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof of  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  Finally, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8, which we restate:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1  such that every graph in Ct(F) has treewidth at most τ(F, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let d and r be the maximum degree and the radius of F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows that  T r  d contains F as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  Let f = f(d, r, 3, t) be as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2 and let  m = m(f, f, t) be as in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Let β(·, ·) be as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' We claim that τ(F, t) =  β(max{m, t + 1}, t) satisfies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Suppose for a contradiction that tw(G) > τ for some  G ∈ Ct(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4, G contains a max{m, t + 1}-block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Consequently, since G is  Kt-free, there are two distinct and non-adjacent vertices a, b ∈ B, and m pairwise internally  disjoint paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' , Pm in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' It follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='3 that G contains T f  f as a  subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Also, since G ∈ Ct(F) ⊆ Ct, G is (K3,3, Kt)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' But now by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='2, G contains  T r  d , and so F, as an induced subgraph, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  ■  References  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Aboulker, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Adler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Kim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Sintiari, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Trotignon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “On the treewidth of even-hole-free  graphs.” European Journal of Combinatorics 98, (2021), 103394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Abrishami, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Alecu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Chudnovsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Hajebi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Spirkl, “Induced subgraphs and tree decomposi-  tions VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Basic obstructions in H-free graphs.” arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='02737, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Abrishami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Chudnovsky, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Dibek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Hajebi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Rzążewski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Spirkl, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Vušković, “Induced  subgraphs and tree decompositions II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Toward walls and their line graphs in graphs of bounded degree.”  arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='01162, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Abrishami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Chudnovsky and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Vušković, “Induced subgraphs and tree decompositions I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Even-hole-  free graphs of bounded degree.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' B, 157 (2022), 144-175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Ajtai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Komlós and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Szemerédi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “A note on Ramsey numbers.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Combinatorial Theory, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' A 29  (1980), 354–360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Bodlaender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “Dynamic programming on graphs with bounded treewidth.” Springer, Berlin, Heidelberg,  (1988), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' 105–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Cameron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' da Silva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Huang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Vušković, “Structure and algorithms for (cap, even hole)-free  graphs.” Discrete Mathematics 341, 2 (2018), 463-473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Chudnovsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Robertson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Seymour, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Thomas, “The strong perfect graph theorem.” Annals of  Math 164 (2006), 51-229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Chudnovsky and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Seymour, “Even-hole-free graphs still have bisimplicial vertices.” arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='10967,  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Chudnovsky and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Seymour, “The three-in-a-tree problem.” Combinatorica 30, 4 (2010): 387-417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Davies, “Vertex-minor-closed classes are χ-bounded.” arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='05069, (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Davies, appeared in an Oberwolfach technical report DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='4171/OWR/2022/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Erde and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Weißauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “A short derivation of the structure theorem for graphs with excluded topological  minors.” SIAM Journal of Discrete Mathematics 33, 3 (2019), 1654–1661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Grohe and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “Structure theorem and isomorphism test for graphs with excluded topological  subgraphs,” SIAM Journal on Computing 44, 1 (2015), 114–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Kierstead and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Penrice, “Radius two trees specify χ-bounded classes.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Graph Theory 18, 2  (1994): 119–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Korhonen, “Grid Induced Minor Theorem for Graphs of Small degree.” arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='13233, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [17] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Lozin and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Razgon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “Tree-width dichotomy.” European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Combinatorics 103 (2022): 103517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Menger, “Zur allgemeinen Kurventheorie.” Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' 10, 1927, 96–115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [19] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Robertson and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Seymour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “Graph minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Excluding a planar graph.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' B, 41  (1) (1996), 92–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Sintiari and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Trotignon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' “(Theta, triangle)-free and (even-hole, K4)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Part 1: Layered  wheels.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Graph Theory 97 (4) (2021), 475-509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content='  [21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}
+page_content=' Trotignon, private communication, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'}