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

##### Abstract

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.

##### Collections

##### Cite this version of the work

Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Sepehr Hajebi, Pawel Rzqzewski, Sophie Spirkl, Kristina Vuskovic
(2024).
Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree.. UWSpace.
http://hdl.handle.net/10012/20107

Other formats

The following license files are associated with this item: