Paired-Domination in Claw-Free Cubic Graphs

  • Authors:

  • Odile Favaron;Michael A. Henning

  • Affiliations:

  • Université de Paris-Sud, Laboratoire de Recherche en Informatique, UMR 8623, 91405, Orsay, France;University of KwaZulu-Natal, School of Mathematics, Statistics & Information Technology, Private Bag X01, 3209, Scottsville, South Africa

  • Venue:
  • Graphs and Combinatorics
  • Year:
  • 2004

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Abstract

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 minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by γpr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K1,3 or K4−e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K1,3,K4−e,C4)-free, then γpr(G)≤3n/8; (ii) if G is claw-free and diamond-free, then γpr(G)≤2n/5; (iii) if G is claw-free, then γpr(G)≤n/2. In all three cases, the extremal graphs are characterized.