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

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 in which this condition is also sufficient, which we call (tw, ω)-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, ω)-bounded. Our results imply that the class of 1-perfectly orientable graphs is (tw, ω)-bounded, answering a question of Brešar, Hartinger, Kos, and Milanič from 2018. We also reveal some further algorithmic implications of (tw, ω)-boundedness related to list k-coloring and clique problems.

Treewidth versus clique number. II. Tree-independence number

“…3. As also shown in [54,55], known approximation algorithms for treewidth (see, e.g., [66]) lead to improved approximations for the clique number of a graph from a (tw, ω)-bounded graph class having a computable exponential binding function. The approximation can be improved further if the binding function is computable and either linear or polynomial.…”

“…Graphs with large cliques necessarily have large treewidth; in chordal graphs the converse holds, too: the treewidth of a chordal graph can only be large if there exists a large clique (see, e.g., [17]). Recently, Dallard, Milanič, and Štorgel [54,55] initiated a systematic study of graph classes in which this sufficient condition for large treewidth-the presence of a large clique-is also necessary, which they call (tw, ω)-bounded.…”

“…Classes of graphs in which all minimal separators are of bounded size were studied in 1999 by Skodinis [107]. In [54,55], the authors characterized, for each of six wellknown graph containment relations (the minor, topological subgraph, subgraph, and their induced variants), the graphs H such that the class of graphs excluding a H with respect to the relation is (tw, ω)-bounded.…”

“…Chaplick and Zeman showed in [38] that for every positive integer k, any (tw, ω)-bounded graph class G having a computable binding function admits linear-time algorithms for the k-Clique and the List k-Coloring problems. Furthermore, as shown by Dallard, Milanič, and Štorgel [54,55], any such class admits a polynomial-time algorithm for the List k-Coloring problem that is robust in the sense of Raghavan and Spinrad [99]: it either solves the problem or determines that the input graph is not in G.…”

“…Understanding what restrictions on the input graph make the problem more tractable has been a subject of investigation for decades, with several recent developments [1,26,34,43,44,72,73,77,98]. Dallard, Milanič, and Štorgel observed in [55] that in all the (tw, ω)-bounded graph classes defined by excluding a single graph H with respect to any of the six considered graph containment relations (the minor, topological subgraph, subgraph, and their induced variants), the MWIS problem problem can be solved in polynomial time, except possibly for the case when H is excluded with respect to the induced minor relation and H is isomorphic to either K − 5 (the complete 5-vertex graph minus an edge), W 4 (a wheel with four spokes, that is, the graph obtained from the 4-vertex cycle by adding to it a universal vertex), or the complete bipartite graph K 2,q where q ≥ 3. (See Section 2 for precise definitions.)…”