Restrained domination in cubic graphs

Johannes H. Hattingh and Ernst J. Joubert ()
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Johannes H. Hattingh: Georgia State University
Ernst J. Joubert: University of Johannesburg

Journal of Combinatorial Optimization, 2011, vol. 22, issue 2, No 4, 166-179

Abstract: Abstract Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ 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 γ r (G)=γ(G).

Keywords: Graph; Cubic graph; Domination; Restrained domination; Upper bound; Lower bound (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10878-009-9281-2

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