Induced subgraphs and tree decompositions V. One neighbor in a hole

Abstract

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions.") So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded teewidth. Sintiari and Trotignon answered this question in negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e. an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether wheels are essential to the construction of layered wheels, that is, whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out be false due to Davies' recent example of graphs with large treewidth and no induced subgraph isomorphic to (fairly small) basic treewidth obstructions, in which every vertex has at most two neighbors in every hole.Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in a counterexample: graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

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

“…• 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]).…”

Induced subgraphs and tree decompositions I. Even-hole-free graphs of bounded degree

“…Therefore, the focus now is to determine which graph classes of unbounded degree have bounded treewidth. Several previous papers in this series have proven that certain hereditary graph classes of unbounded degree have bounded treewidth; see [1,4]. Graph classes in which treewidth is bounded by a logarithmic function of the number of vertices have also been studied ( [3,7]).…”

“…For a graph H, let S H ⊆ V (H) denote the set of vertices of H of degree at least three. By Theorem 7.1 of [1], there exists a constant w(t, ∆) such that for every graph H, if H has treewidth at least w(t, ∆) and S H has maximum degree at most ∆, then H contains a subdivision of the (t × t)-wall or the line graph of a subdivision of the (t × t)-wall as an induced subgraph. We apply this theorem to G. By definition of strongly 2-bipartite, it follows that S G is a partite set of a bipartition of G, and thus has maximum degree 0.…”

“…We apply this theorem to G. By definition of strongly 2-bipartite, it follows that S G is a partite set of a bipartition of G, and thus has maximum degree 0. Let f (t) = w(t, 0), where w is as in Theorem 7.1 of [1]. Now, since G is t-clean, it follows that tw(G) ≤ f (t).…”