Mixed Dominating Set: A Parameterized Perspective

Abstract

In the mixed dominating set (mds) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether there exists a set $S \subseteq V(G) \cup E(G)$ of cardinality at most k such that every element $x \in (V(G) \cup E(G)) \setminus S$ is either adjacent to or incident with an element of S. We show that mds can be solved in time ${7.465^k n^{\mathcal {O}(1)}}$ on general graphs, and in time $2^{\mathcal {O}(\sqrt{k})} n^{\mathcal {O}(1)}$ on planar graphs. We complement this result by showing that mds does not admit an algorithm with running time $2^{o(k)} n^{\mathcal {O}(1)}$ unless the Exponential Time Hypothesis (ETH) fails, and that it does not admit a polynomial kernel unless coNP $\subseteq \mathsf{NP / poly}$. In addition, we provide an algorithm which, given a graph G together with a tree decomposition of width $\mathsf{tw}$, solves mds in time $6^{\mathsf{tw}} n^{\mathcal {O}(1)}$. We finally show that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time $\mathcal {O}((2-\epsilon )^{\mathsf{tw}(G)} n^{\mathcal {O}(1)})$ for any $\epsilon >0$, where $\mathsf{tw}(G)$ is the tree-width of G.

The research leading to these results have received funding from the European Research Council via ERC Advanced Investigator Grant 267959.