Treewidth versus clique number. I. Graph classes with a forbidden structure

TL;DR

This paper characterizes graph classes with bounded treewidth based on forbidden structures, revealing new algorithmic applications and solving open problems related to coloring and clique problems.

Contribution

It provides a complete characterization of $(tw, ext{}\omega)$-bounded classes for six graph containment relations, and shows that certain classes like $1$-perfectly orientable graphs are $(tw, ext{}\omega)$-bounded.

Findings

Characterization of $(tw, ext{ }\omega)$-bounded classes for six containment relations.

Linear-time algorithms for $k$-coloring $1$-perfectly orientable graphs.

Polynomial-time solvability of maximum weight independent set in classes excluding a fixed star as an induced minor.

Abstract

Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $(tw,\omega)$-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and $k$-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs $H$ for which the class of graphs excluding $H$ is $(tw,\omega)$-bounded. Our results yield an infinite family of $\chi$-bounded induced-minor-closed graph classes and imply that the class of $1$-perfectly orientable graphs is $(tw,\omega)$-bounded, leading to linear-time algorithms for $k$-coloring $1$-perfectly orientable graphs for every fixed~$k$. This answers a question of Bre\v sar, Hartinger, Kos, and Milani{\v c} from 2018 and one of Beisegel, Chudnovsky, Gurvich, Milani{\v c}, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of $(tw,\omega)$-boundedness related to list $k$-coloring and clique problems. In addition, we propose a question about the complexity of the maximum weight independent set problem in $(tw,\omega)$-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor.