Paired-domination in inflated graphs

  • Authors:

  • Liying Kang;Moo Young Sohn;T. C. E. Cheng

  • Affiliations:

  • Department of Mathematics, Shanghai University, Shanghai 200436, China;Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea;Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2004

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Abstract

The inflation G1 of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique Kd. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number λp(G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G) ≥ 2, then n(G) ≤ λp(G1) ≤ 4m(G)/[δ(G) + 1], and the equality λp(G1)=n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.