Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

Abstract

A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family H of graphs, we say a graph G is H-free if no induced subgraph of G is isomorphic to a member of H. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant c for which every (theta, triangle)-free graph G has treewidth at most c log(|V (G)|). A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in |V (G)| for every graph G excluding the so-called three-path-configurations as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and k-Coloring (for fixed k) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.

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

“…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]).…”

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

“…Theorem 1.4 is the first result of this series that gives a constant bound on treewidth in a class of graphs with arbitrary maximum degree; the previous results have either obtained a constant bound on treewidth in graph classes with bounded degree ( [2], [4]), or given a logarithmic bound on treewidth in graph classes with bounded clique number ( [3]). Bounded treewidth results for similar graph classes were also proved in [7].…”

“…Because of the way the decompositions corresponding to star cutsets are constructed, we obtain an induced subgraph β of G such that either β is wheel-free or β has a balanced separator. To prove that β has a balanced separator if it is not wheel-free, we use degeneracy to bound the degree of vertices which are wheel centers in β (a similar technique was used in [3]). In either case, β has bounded treewidth, and so G has bounded treewidth.…”