Induced subgraphs and tree decompositions XVII. Anticomplete sets of large treewidth

TL;DR

The paper proves that large treewidth graphs necessarily contain two large anticomplete sets with large treewidth unless they contain specific highly structured subgraphs, extending understanding of graph decomposition.

Contribution

It establishes a new structural property of large treewidth graphs involving anticomplete sets, identifying exceptions as specific structured graphs.

Findings

Large treewidth graphs contain two large anticomplete sets with large treewidth.

Exceptions are characterized as complete graphs, complete bipartite graphs, or interrupted s-constellations.

The result extends the theory of graph decompositions and treewidth.

Abstract

Two sets $X, Y$ of vertices in a graph $G$ are "anticomplete" if $X\cap Y=\varnothing$ and there is no edge in $G$ with an end in $X$ and an end in $Y$. We prove that every graph $G$ of sufficiently large treewidth contains two anticomplete sets of vertices each inducing a subgraph of large treewidth unless $G$ contains, as an induced subgraph, a highly structured graph of large treewidth that is an obvious counterexample to this statement. These are: complete graphs, complete bipartite graphs and "interrupted $s$-constellations." The latter is a slightly adjusted version of a well-known construction by Bonamy et al.