Induced subgraphs and tree decompositions IV. (Even hole, diamond, pyramid)-free graphs

Abstract

A hole in a graph G is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph K4 by removing an edge. A pyramid is a graph consisting of a triangle called the base, a vertex called the apex, and three internally disjoint paths starting at the apex and disjoint otherwise, each joining the apex to a vertex of the base. For a family H of graphs, we say a graph G is Hfree if no induced subgraph of G is isomorphic to a member of H. Cameron, da Silva, Huang, and Vušković proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, K4)-free graphs of arbitrarily large treewidth, and observing induced diamonds all over their construction, they conjectured that (even hole, diamond, K4)-free graphs have bounded treewidth. We approach this conjecture by showing that for every t, (even hole, pyramid, diamond, Kt)-free graphs have bounded treewidth. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses "non-crossing decompositions" methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of "non-crossing decompositions" to prove bounded treewidth in a graph class of unbounded maximum degree.

Grid Induced Minor Theorem for Graphs of Small Degree

“…In addition to [1], this conjecture has been explicitly mentioned by Abrishami, Chudnovsky, Dibek, Hajebi, Rzazewski, Spirkl, and Vušković in three articles of their "Induced subgraphs and tree decompositions" series [2,3,4,5] and the main results of the first two articles of this series are special cases of this conjecture.…”

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]).…”