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 K t K t and K t S t 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 (K 3 S 3 )-free chordal graphs; and (K t S t )-free chordal graphs have mim-width at most t − 1,• permutation graphs are (K 3 K 3 )-free co-comparability graphs; and (K t K t )-free cocomparability 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 n O(t) on (K t S t )-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 = N P .We generalize these ideas to bigger graph classes. We introduce a new width parameter simwidth, of stronger modelling 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 K t K t and K t S t 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.

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

“…Furthermore, for each fixed integer k ≥ 1, the k-domination and total k-domination problems are solvable in time O(n 6k+4 ) in the class of interval graphs where n is the order of the input graph. This follows from recent results due to Kang et al [41], building on previous works by Bui-Xuan et al [8] and Belmonte and Vatshelle [3]. In fact, Kang et al studied a more general class of problems, called (ρ, σ)-domination problems, and showed that every such problem can be solved in time O(n 6d+4 ) in the class of n-vertex interval graphs, where d is a parameter associated to the problem (see Corollary 3.2 in [41] and the paragraph following it).…”

“…The approach of Kang et al [41], which implies that k-domination and total k-domination are solvable in time O(|V (G)| 6k+4 ) in the class of interval graphs also works for the weighted versions of the problems, where each vertex u ∈ V (G) is equipped with a non-negative cost c(u) and the task is to find a (usual or total) k-dominating set of G of minimum total cost. For both families of problems, our approach can also be easily adapted to the weighed case.…”

“…These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work of Kang et al (2017) that these two families of problems are solvable in time O(|V (G)| 6k+4 ) in the class of interval graphs. We develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs, by means of reduction to a single shortest path computation in a derived directed acyclic graph with O(|V (G)| 2k ) nodes and O(|V (G)| 4k ) arcs.…”

Bounding the mim‐width of hereditary graph classes

“…Kang et al [39] also proved that the classes of chordal graphs, circle graphs and co‐comparability graphs have unbounded mim‐width; for the latter two classes, this was shown independently by Mengel [43]. Vatshelle [49] and Brault‐Baron et al [13] showed the same for grids and chordal bipartite graphs, respectively, whereas Mengel [43] proved that strongly chordal split graphs have unbounded mim‐width.…”

“…Let be the graph obtained from by adding a perfect matching, and let be the graph obtained from by removing all the edges in one of the complete graphs (see Section 2 for undefined notation). Kang et al [39] showed that for any integer , there is a polynomial‐time algorithm for computing a branch decomposition of mim‐width at most when the input is restricted to ‐free chordal graphs, which generalize interval graphs, or ‐free co‐comparability graphs, which generalize permutation graphs. Hence, in particular, all these classes have bounded mim‐width.…”