Complexity of distance paired-domination problem in graphs

Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matching. Given a graph G = (V , E) and a fixed integer k > 0, the Min Distance k-Paired-Dom Set problem is to find a minimum cardinality distance k-paired-dominating set of G. In this paper, we show that the decision version of Min Distance k-Paired-Dom Set is NP-complete for undirected path graphs. This strengthens the complexity of decision version ofMin Distance k-Paired-Dom Set problem in chordal graphs. We show that for a given graph G, unless NP ⊆ DTIME (nO(log log n)), Min Distance k-Paired-Dom Set problem cannot be approximated within a factor of (1 − ε) ln n for any ε > 0, where n is the number of vertices in G. We also show thatMin Distance k-PairedDom Set problem is APX-complete for graphs with degree bounded by 3. On the positive side, we present a linear time algorithm to compute the minimum cardinality of a distance k-paired-dominating set of a strongly chordal graph G if a strong elimination ordering of G is provided. We show that for a given graph G, Min Distance k-Paired-Dom Set problem can be approximated with an approximation factor of 1+ ln 2+ k · ln(∆(G)), where ∆(G) denotes the maximum degree of G. © 2012 Elsevier B.V. All rights reserved.