On the Complexity of Minimum Membership Dominating Set

Abstract

In this paper, we study a variant of Domination called Minimum Membership Dominating Set, in short MMDS. The input to the problem is a graph G and an integer k (which is the membership parameter). The goal is to compute a set $S\subseteq V(G)$ such that for each $v\in V(G)$, $1\le |N[v]\cap S|\le k$. Notice that there is no requirement on the size of S. We extend the study on this problem from the parameterized complexity perspective. The following are the results of this paper.

• Agrawal et al. (Algorithmica 2023) showed that MMDS is W[1]-hard parameterized by pathwidth of the input graph. They asked as open questions if MMDS is FPT when parameterized by maximum degree, distance to bounded degree graphs or maximum number of leaves in a spanning tree.

  We show that MMDS is NP-hard even on graphs with maximum degree three. This answers the first two questions in negative. We consider the parameter distance to disjoint paths that generalizes the maximum number of leaves in a spanning tree and show that the problem is FPT.

• Recently Sangam et al. (2024) showed that MMDS is FPT parameterized by the combined parameters distance to cluster and membership. However the running time of the algorithm is triple exponential. We design a single exponential FPT algorithm parameterized by distance to cluster alone. We also obtain an FPT algorithm parameterized by distance to co-cluster.

• We show that MMDS can be solved in time $k^{5\textsf{cw}}n^{O(1)}$ where $\mathsf cw$ is the cliquewidth of the input graph. This implies a polynomial time algorithm for graphs of bounded cliquewidth. In particular distance hereditary graphs that have cliquewidth at most three, thereby resolving an open question asked by Sangam et al. in the above paper.