Complexity of distance paired-domination problem in graphs

  • Authors:

  • Gerard J. Chang;B. S. Panda;D. Pradhan

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan and Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan and National Center for T ...;Computer Science and Application Group, Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110 016, India;Department of Computer Science and Automation, Indian Institute of Science, Bangalore-560 012, India

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2012

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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 d"G(u,v)@?k, where d"G(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 k0, 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(n^O^(^l^o^g^l^o^g^n^)), Min Distancek-Paired-Dom Set problem cannot be approximated within a factor of (1-@e)lnn for any @e0, 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(@D(G)), where @D(G) denotes the maximum degree of G.