Sim-width and induced minors

We introduce a new graph width parameter, called special induced matching width, shortly sim-width, which does not increase when taking induced minors. For a vertex partition (A,B) of a graphG, this parameter is based on the maximum size of an induced matching {a1b1, . . . , ambm} in G where a1, . . . , am ∈ A and b1, . . . , bm ∈ B. Classes of graphs of bounded sim-width are much wider than classes of bounded tree-width, rank-width, or mim-width. As examples, we show that chordal graphs and co-comparability graphs have sim-width at most 1, while they have unbounded value for the other three parameters. In this paper, we obtain general algorithmic results on graphs of bounded sim-width by further excluding certain graphs as induced minors. A t-matching complete graph is a graph that consists of two vertex sets {v1, . . . , vt} and {w1, . . . , wt} such that {v1, . . . , vt} is a clique, and between the two sets, {v1w1, . . . , vtwt} is an induced matching. We prove that for positive integers w and t, a large class of domination and partitioning problems, including the Minimum Dominating Set problem, can be solved in time nO(wt ) on n-vertex graphs of sim-width at most w and having no induced minor isomorphic to a t-matching complete graph, when the decomposition tree is given. To prove it, we show that such graphs have mim-width at most (4w + 2)t. In this way, we generate infinite nontrivial classes of graphs that are closed under induced minors, but not under minors, and have general algorithmic applications. For chordal graphs and co-comparability graphs, we provide polynomial-time algorithms to obtain decomposition trees certifying that their sim-width are at most 1. Note that Minimum Dominating Set is NP-complete on chordal graphs.