Global total k-domination: Approximation and hardness results

Abstract

A subset D⊆V of a graph G=(V,E) is called a global total k-dominating set of G if D is a total k-dominating set of both G and the complement G‾ of G. The Minimum Global Totalk-Domination problem is to find a global total k-dominating set of minimum cardinality of the input graph G and Decide Global Totalk-Domination problem is the decision version of Minimum Global Totalk-Domination problem. The Decide Global Totalk-Domination problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of Decide Global Totalk-Domination problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, star-convex bipartite graphs and doubly chordal graphs. On the positive side, we give a polynomial time algorithm for the Minimum Global Totalk-Domination problem for chordal bipartite graphs, which is a subclass of bipartite graphs. We propose a 2(1+ln⁡|V|)-approximation algorithm for the Minimum Global Totalk-Domination problem for any graph. We show that Minimum Global Totalk-Domination problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless P=NP for any integer k≥1. We further show that for bipartite graphs, Minimum Global Totalk-Domination problem cannot be approximated within (16−ϵ)ln⁡|V| for any ϵ>0 unless P=NP for any k≥1. Finally, we show that the Minimum Global Totalk-Domination problem is APX-complete for bounded degree bipartite graphs for any fixed integer k≥1.