Parameterized Complexity of Minimum Membership Dominating Set

Given a graph G= (V, E) and an integer k, the Minimum Membership Dominating Set (MMDS) problem seeks to find a dominating set S⊆ V of G such that for each v∈ V , | N[v] ∩ S| is at most k. We investigate the parameterized complexity of the problem and obtain the following results for the MMDS problem. First, we show that the MMDS problem is NP-hard even on planar bipartite graphs. Next, we show that the MMDS problem is W[1]-hard for the parameter pathwidth (and thus, for treewidth) of the input graph. Then, for split graphs, we show that the MMDS problem is W[2]-hard for the parameter k. Further, we complement the pathwidth lower bound by an FPT algorithm when parameterized by the vertex cover number of input graph. In particular, we design a 2 ^O(vc)| V| ^O(1) time algorithm for the MMDS problem where vc is the vertex cover number of the input graph. Finally, we show that the running time lower bound based on ETH is tight for the vertex cover parameter.