# 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 about MMDS: W[1]-hardness of the problem parameterized by the pathwidth (and thus, treewidth) of the input graph. W[1]-hardness parameterized by k on split graphs. An algorithm running in time 2^𝒪(vc) |V|^𝒪(1), where vc is the size of a minimum-sized vertex cover of the input graph. An ETH-based lower bound showing that the algorithm mentioned in the previous item is optimal.

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