Differentiating-total domination: Approximation and hardness results

Abstract

A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V is adjacent to at least one vertex of the set D. A total dominating set D of G is a differentiating-total dominating set of G if for every pair of distinct vertices u,v∈V, NG[u]∩D≠NG[v]∩D. Given a graph G, Minimum Differentiating-Total Domination is to find a differentiating-total dominating set of minimum cardinality of G and Decide Differentiating-Total Domination is the decision version of Minimum Differentiating-Total Domination. In this paper, we initiate the algorithmic study of Minimum Differentiating-Total Domination. We show that Decide Differentiating-Total Domination is NP-complete for chordal graphs, chordal bipartite graphs, star-convex bipartite graphs, and planar graphs. On the positive side, we propose an O(log⁡Δ) factor approximation algorithm for Minimum Differentiating-Total Domination for any graph G with maximum degree Δ. We match the above upper bound by showing that for general graphs as well as for bipartite graphs Minimum Differentiating-Total Domination cannot be approximated within a factor of (12−ε)ln⁡|V| for any ε>0 unless P= NP. Finally, we show that Minimum Differentiating-Total Domination is APX-complete for bounded degree bipartite graphs.