Complexity and Algorithms for Semipaired Domination in Graphs

Abstract

For a graph $G=(V,E)$ with no isolated vertices, a set $D\subseteq V$ is called a semipaired dominating set of G if (i) D is a dominating set of G, and (ii) D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted by $\gamma _{pr2}(G)$. The Minimum Semipaired Domination problem is to find a semipaired dominating set of G of cardinality $\gamma _{pr2}(G)$. In this paper, we initiate the algorithmic study of the Minimum Semipaired Domination problem. We show that the decision version of the Minimum Semipaired Domination problem is NP-complete for bipartite graphs and chordal graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs. We also propose a $1+\ln (2\varDelta +2)$-approximation algorithm for the Minimum Semipaired Domination problem, where $\varDelta$ denotes the maximum degree of the graph and show that the Minimum Semipaired Domination problem cannot be approximated within $(1-\epsilon ) \ln |V|$ for any $\epsilon > 0$ unless P = NP.

Research of Michael A. Henning was supported in part by the University of Johannesburg.