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.

In the second part, we generalize these results to bigger classes. We introduce a new width parameter sim-width, special induced matching-width, by making only a small change in the definition of mim-width. We prove that chordal and co-comparability graphs have sim-width at most 1. Since Dominating Set is NP-complete on chordal graphs, an XP algorithm parameterized only by sim-width would imply P = NP. Therefore, to apply the algorithms for domination-type problems mentioned above, we parameterize by both sim-width w and a further parameter t, which is the smallest value such that the input has no induced minor isomorphic to $K_t \boxminus S_t$ or $K_t \boxminus S_t$. We show that such graphs have mim-width at most $8(w+1)t^3$ and that the resulting algorithms for domination-type problems have runtime $n^{\mathcal {O}(wt^3)}$, when the decomposition tree is given.

The first author is supported by TJ Park Science Fellowship of POSCO TJ Park Foundation. The second author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC consolidator grant DISTRUCT, agreement No 648527).