Induced subgraphs and tree decompositions VIII. Excluding a forest in (theta, prism)-free graphs

Abrishami, Tara; Alecu, Bogdan; Chudnovsky, Maria; Hajebi, Sepehr; Spirkl, Sophie

doi:10.48550/arxiv.2301.02138

Cited by 2 publications

(5 citation statements)

References 19 publications

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“…By Lemma 4.5, there exists a nonempty path P v ⊆ P on at most 2(d − 1)(q − 2) vertices such that N (v) ∩ V (P ) ⊆ V (P v ) and each endpoint of P v is adjacent to v. Let p 1 and p 2 be the two endpoints of P v (possibly p 1 = p 2 ) and let P ′ 1 and P ′ 2 be two subpaths of P such that the paths P ′ 1 , P v , P ′ 2 are pairwise edge-disjoint and with union P , and p i ∈ V (P ′ i ) for i ∈ {1, 2}. For each i ∈ [2], if P ′ i has at most q vertices let P i := P ′ i , otherwise let P i be the subpath of P ′ i on q + 1 vertices that contains p i . Let P H := P 1 p 1 P v p 2 P 2 .…”

Section: Proposition 21 (Folklore) Let G Be a Graph And H A Minor Of ...mentioning

confidence: 99%

“…This parameter is called tree-independence number and is denoted by tree-α(G). 2 It is defined similarly as treewidth, via tree decompositions, except that the measure of quality of a tree decomposition is changed; instead of measuring the maximum cardinality of a bag, what matters is the independence number of the decomposition, defined as the maximum cardinality of an independent set contained in a bag. For problems related to independent sets, this measure allows for the development of polynomial-time dynamic programming algorithms (see [39,70]).…”

Section: Introductionmentioning

confidence: 99%

“…• Over a series of papers, Abrishami et al recently studied induced obstructions to bounded treewidth (see [2,3,5,[7][8][9][10]12]). In particular, they showed that the class of (even hole, diamond, pyramid)-free graphs is (tw, ω)-bounded (see [9]).…”

Section: Introductionmentioning

confidence: 99%

“…The result assumes that the graph is equipped with a tree decomposition of bounded width. As shown by Bodlaender[20], such a tree decomposition can be computed in linear time 2. Yolov called it α-treewidth in[70].…”

mentioning

confidence: 99%

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Treewidth versus Clique Number. I. Graph Classes with a Forbidden Structure

Dallard¹,

Milanič²,

Štorgel³

2021

SIAM J. Discrete Math.

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Many recent works address the question of characterizing induced obstructions to bounded treewidth. In 2022, Lozin and Razgon completely answered this question for graph classes defined by finitely many forbidden induced subgraphs. Their result also implies a characterization of graph classes defined by finitely many forbidden induced subgraphs that are (tw, ω)-bounded, that is, treewidth can only be large due to the presence of a large clique. This condition is known to be satisfied for any graph class with bounded tree-independence number, a graph parameter introduced independently by Yolov in 2018 and by Dallard, Milanič, and Štorgel in 2024. Dallard et al. conjectured that (tw, ω)-boundedness is actually equivalent to bounded tree-independence number. We address this conjecture in the context of graph classes defined by finitely many forbidden induced subgraphs and prove it for the case of graph classes excluding an induced star. We also prove it for subclasses of the class of line graphs, determine the exact values of the tree-independence numbers of line graphs of complete graphs and line graphs of complete bipartite graphs, and characterize the tree-independence number of P 4 -free graphs, which implies a linear-time algorithm for its computation. Applying the algorithmic framework provided in a previous paper of the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes.

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Section: Proposition 21 (Folklore) Let G Be a Graph And H A Minor Of ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Treewidth versus Clique Number. I. Graph Classes with a Forbidden Structure

Dallard¹,

Milanič²,

Štorgel³

2021

SIAM J. Discrete Math.

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“…Let f (k, n) := |V (T g(max{k,n}) )| − 1 where g is from Lemma 15. 3 By Theorem 4, G contains a minor model (X v : v ∈ V (T g(max{k,n}) )) of T g(max{k,n}) . Let G be the induced minor of G obtained from contracting each of the X v 's.…”

Section: Lemma 15 ([9 20]mentioning

confidence: 99%

Induced Subgraphs and Path Decompositions

Hickingbotham

2023

Electron. J. Combin.

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A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Similarly, we show that every graph excluding a fixed minor and with sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Finally, we present a characterisation for when a hereditary class defined by a finite set of forbidden induced subgraphs has bounded pathwidth.

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Induced subgraphs and tree decompositions VIII. Excluding a forest in (theta, prism)-free graphs

Abstract: 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.

Cited by 2 publications

References 19 publications

Treewidth versus Clique Number. I. Graph Classes with a Forbidden Structure

Treewidth versus Clique Number. I. Graph Classes with a Forbidden Structure

Induced Subgraphs and Path Decompositions

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