On the Hardness of Generalized Domination Problems Parameterized by Mim-Width

Abstract

For nonempty ?, ? ? ?, a vertex set S in a graph G is a (?, ?)-dominating set if for all v ? S, |N(v) ? S| ? ?, and for all v ? V(G) ? S, |N(v) ? S| ? ?. The Min/Max (?,?)-Dominating Set problems ask, given a graph G and an integer k, whether G contains a (?, ?)-dominating set of size at most k and at least k, respectively. This framework captures many well-studied graph problems related to independence and domination. Bui-Xuan, Telle, and Vatshelle [TCS 2013] showed that for finite or co-finite ? and ?, the Min/Max (?,?)-Dominating Set problems are solvable in XP time parameterized by the mim-width of a given branch decomposition of the input graph. In this work we consider the parameterized complexity of these problems and obtain the following: For minimization problems, we complete several scattered W[1]-hardness results in the literature to a full dichotomoy into polynomial-time solvable and W[1]-hard cases, and for maximization problems we obtain the same result under the additional restriction that ? and ? are finite sets. All W[1]-hard cases hold assuming that a linear branch decomposition of bounded mim-width is given, and with the solution size being an additional part of the parameter. Furthermore, for all W[1]-hard cases we also rule out f(w)n^o(w/log w)-time algorithms assuming the Exponential Time Hypothesis, where f is any computable function, n is the number of vertices and w the mim-width of the given linear branch decomposition of the input graph