A Width Parameter Useful for Chordal and Co-comparability Graphs

Abstract

In 2013 Belmonte and Vatshelle used mim-width, a graph parameter bounded on interval graphs and permutation graphs that strictly generalizes clique-width, to explain existing algorithms for many domination-type problems, known as LC-VSVP problems. We focus on chordal graphs and co-comparability graphs, that strictly contain interval graphs and permutation graphs respectively. First, we show that mim-width is unbounded on these classes, thereby settling an open problem from 2012. Then, we introduce two graphs \(K_t \boxminus K_t\) and \(K_t \boxminus S_t\) to restrict these graph classes, 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 \((K_t \boxminus S_t)\)-free chordal graphs have mim-width at most \(t-1\), and \((K_t \boxminus K_t)\)-free co-comparability graphs have mim-width at most \(t-1\). From this, we obtain several algorithmic consequences, for instance, while Dominating Set is NP-complete on chordal graphs, it can be solved in time \(\mathcal {O}(n^{t})\) on chordal graphs where t is the maximum among induced subgraphs \(K_t \boxminus S_t\) in the given graph. We also show that classes restricted in this way have unbounded rank-width which validates our approach.