Tree-width dichotomy

Lozin, Vadim; Razgon, Igor

doi:10.48550/arxiv.2012.01115

Cited by 1 publication

(4 citation statements)

References 11 publications

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“…From the definition of a (k, t)-creature, it follows directly that H and P satisfy the above three bullets. This proves (10).…”

Section: Proof Suppose Not Letsupporting

confidence: 53%

“…Given a graph F , the line graph L(F ) of F is the graph with vertex set E(F ), such that two vertices of L(F ) are adjacent if the corresponding edges of G share an end. Theorem 1.2 ( [10]). Let F be finite family of graphs.…”

Section: Theorem 11 ([12]mentioning

confidence: 99%

“…Then there exists a graph G of maximum degree at most ∆, containing a (k, t)-creature and with no induced subgraph isomorphic to either a subdivision of T or the line graph of a subdivision of T . (10) We may choose H and P with the following specifications.…”

Section: Proof Suppose Not Letmentioning

confidence: 99%

“…We choose H and P satisfying (10) and with |H| as small as possible. For every P ∈ P, let v P be as in the third bullet of (10). Let A = {v P : P ∈ P} and J = H \ A.…”

Section: Proof Suppose Not Letmentioning

confidence: 99%

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Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

Abrishami¹,

Chudnovsky²,

Dibek³

et al. 2021

Preprint

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This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of the (k × k)-wall or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.1. For t ≥ 2, a t-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all k, t and ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every ∆ and subcubic subdivided caterpillar T , every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

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“…From the definition of a (k, t)-creature, it follows directly that H and P satisfy the above three bullets. This proves (10).…”

Section: Proof Suppose Not Letsupporting

confidence: 53%

Section: Theorem 11 ([12]mentioning

confidence: 99%

Section: Proof Suppose Not Letmentioning

confidence: 99%

“…We choose H and P satisfying (10) and with |H| as small as possible. For every P ∈ P, let v P be as in the third bullet of (10). Let A = {v P : P ∈ P} and J = H \ A.…”

Section: Proof Suppose Not Letmentioning

confidence: 99%

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