Semitotal domination on AT-Free graphs and circle graphs

Abstract

For a graph G = (V,E) with no isolated vertices, a set D ⊆ V is called a semitotal dominating set of G if (i) D is a dominating set of G, and (ii) every vertex in D has another vertex in D at a distance at most two. The minimum cardinality of a semitotal dominating set of G is called the semitotoal domination number of G, and is denoted by γt2(G). The Minimum Semitotal Domination problem is to find a semitotal dominating set of G of cardinality γt2(G). In this paper, we present some algorithmic results on Semitotal Domination. We show that the decision version of the Minimum Semitotal Domination problem is NP-complete for circle graphs. On the positive side, we show that the Minimum Semitotal Domination problem is polynomial-time solvable for AT-free graphs. We also prove that the Minimum Semitotal Domination for AT-free graphs can be approximated within approximation ratio of 3 in linear-time. Our results answer the open questions posed by Galby et al. in their recent paper.