TY - JOUR
T1 - Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree
AU - Abrishami, Tara
AU - Chudnovsky, Maria
AU - Dibek, Cemil
AU - Hajebi, Sepehr
AU - Rzążewski, Paweł
AU - Spirkl, Sophie
AU - Vušković, Kristina
N1 - Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2024/1
Y1 - 2024/1
N2 - 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 Δ, every graph with maximum degree at most Δ 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 vertex-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 vertex-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 Δ, every graph with maximum degree at most Δ 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 Δ and subcubic subdivided caterpillar T, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.
AB - 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 Δ, every graph with maximum degree at most Δ 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 vertex-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 vertex-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 Δ, every graph with maximum degree at most Δ 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 Δ and subcubic subdivided caterpillar T, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.
KW - Induced Subgraph
KW - Tree decomposition
KW - Treewidth
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U2 - 10.1016/j.jctb.2023.10.005
DO - 10.1016/j.jctb.2023.10.005
M3 - Article
AN - SCOPUS:85175004566
SN - 0095-8956
VL - 164
SP - 371
EP - 403
JO - Journal of Combinatorial Theory. Series B
JF - Journal of Combinatorial Theory. Series B
ER -