Domination number in graphs with minimum degree two |
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| Authors: | Er Fang Shan Moo Young Sohn Xu Dong Yuan Michael A. Henning |
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| Affiliation: | (1) Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China;(2) Department of Applied Mathematics, Changwon National University, 641-773 Changwon, Korea;(3) Department of Mathematics, Guangxi Normal University, Guilin, 541004, P. R. China;(4) School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa |
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| Abstract: | A set D of vertices of a graph G = (V,E) is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed’s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most (3n+|V 2|)/8, where V 2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number r k (G, γ) ≤ (3n+5k)/8 for all graphs of order n with minimum degree at least 3. This paper is dedicated to the memory of Professor Xudong Yuan. |
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| Keywords: | graph dominating set domination number restricted domination number |
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