A width parameter useful for chordal and co-comparability graphs

Abstract

Belmonte and Vatshelle (TCS 2013) used mim-width, a graph width parameter bounded on interval graphs and permutation graphs, to explain existing algorithms for many domination-type problems on those graph classes. We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs Kt⊟Kt and Kt⊟St are graphs obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that:•interval graphs are (K3⊟S3)-free chordal graphs; and (Kt⊟St)-free chordal graphs have mim-width at most t−1,•permutation graphs are (K3⊟K3)-free co-comparability graphs; and (Kt⊟Kt)-free co-comparability graphs have mim-width at most t−1,•chordal graphs and co-comparability graphs have unbounded mim-width in general. We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time nO(t) on (Kt⊟St)-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless P=NP.We generalize these ideas to bigger graph classes. We introduce a new width parameter sim-width, of stronger modeling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding Kt⊟Kt and Kt⊟St as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width.