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.