Complexity and algorithms for semipaired domination in graphs

Abstract

For a graph G = (V, E) with no isolated vertices, a set D ⊆ 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 γpr2(G). The MINIMUM SEMIPAIRED DOMINATION problem is to find a semipaired dominating set of G of cardinality γpr2(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 split graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs and trees. We also propose a 1 + ln(2∆ + 2)-approximation algorithm for the MINIMUM SEMIPAIRED DOMINATION problem, where ∆ denote the maximum degree of the graph and show that the MINIMUM SEMIPAIRED DOMINATION problem cannot be approximated within (1 − ) ln |V | for any > 0 unless NP ⊆ DTIME(|V | O(log log |V |) ).