Algorithmic aspects of semitotal domination in graphs

Abstract

For a graph G=(V,E), a set D⊆V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D. The Minimum Semitotal Domination problem is to find a semitotal dominating set of minimum cardinality. Given a graph G and a positive integer k, the Semitotal Domination Decision problem is to decide whether G has a semitotal dominating set of cardinality at most k. The Semitotal Domination Decision problem is known to be NP-complete for general graphs. In this paper, we show that the Semitotal Domination Decision problem remains NP-complete for planar graphs, split graphs and chordal bipartite graphs. We give a polynomial time algorithm to solve the Minimum Semitotal Domination problem in interval graphs. We show that the Minimum Semitotal Domination problem in a graph with maximum degree Δ admits an approximation algorithm that achieves the approximation ratio of 2+3ln⁡(Δ+1), showing that the problem is in the class log-APX. We also show that the Minimum Semitotal Domination problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless NP ⊆ DTIME (|V|O(log⁡log⁡|V|)). Finally, we prove that the Minimum Semitotal Domination problem is APX-complete for bipartite graphs with maximum degree 4.