Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Abstract

We continue the study of (tw, ω)-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Maximum Independent Set and related problems.In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a minmax graph invariant related to tree decompositions. Bounded tree-independence number implies both (tw, ω)-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Packing problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In particular, this implies polynomial-time solvability of the Maximum Weight Independent Set problem.In this paper, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a K 5 minus an edge or the 4-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most 3 vertices, while excluding a complete bipartite graph K 2,q implies the existence of a tree decomposition with independence number at most 2(q − 1).These results are obtained using a variety of tools, including ℓ-refined tree decompositions, SPQR trees, and potential maximal cliques, and actually show that in the bounded cases identified in this work, one can also compute tree decompositions with bounded independence number efficiently. Applying the algorithmic framework provided by the previous paper in the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. In particular, these results apply to the class of 1-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.

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.…”

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In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they call (tw, ω)bounded. The family of (tw, ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size.

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