Restrained domination in cubic graphs

  • Authors:

  • Johannes H. Hattingh;Ernst J. Joubert

  • Affiliations:

  • Department of Mathematics and Statistics, Georgia State University, Atlanta, USA 30303-3083;Department of Mathematics, University of Johannesburg, Auckland Park, South Africa 2006

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2011

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Abstract

Let G=(V,E) be a graph. A set S驴V is a restrained dominating set if every vertex in V驴S is adjacent to a vertex in S and to a vertex in V驴S. The restrained domination number of G, denoted 驴 r (G), is the smallest cardinality of a restrained dominating set of G. A graph G is said to be cubic if every vertex has degree three. In this paper, we study restrained domination in cubic graphs. We show that if G is a cubic graph of order n, then $\gamma_{r}(G)\geq \frac{n}{4}$ , and characterize the extremal graphs achieving this lower bound. Furthermore, we show that if G is a cubic graph of order n, then $\gamma _{r}(G)\leq \frac{5n}{11}.$ Lastly, we show that if G is a claw-free cubic graph, then 驴 r (G)=驴(G).