Complexity of distance paired-domination problem in graphs

Abstract

Suppose G=(V,E) is a simple graph and k is a fixed positive integer. A subset D⊆V is a distancek-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 distancek-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 Distancek-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 Distancek-Paired-Dom Set is NP-complete for undirected path graphs. This strengthens the complexity of decision version of Min Distancek-Paired-Dom Set problem in chordal graphs. We show that for a given graph G, unless NP⊆DTIME(nO(loglogn)), Min Distancek-Paired-Dom Set problem cannot be approximated within a factor of (1−ε)lnn for any ε>0, where n is the number of vertices in G. We also show that Min Distancek-Paired-Dom 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 Distancek-Paired-Dom Set problem can be approximated with an approximation factor of 1+ln2+k⋅ln(Δ(G)), where Δ(G) denotes the maximum degree of G.