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.
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This paper is dedicated to the memory of Professor Xudong Yuan.
This research is partially supported by Korea Research Foundation Grant (KRF-2002-015-CP0050); the National Natural Science Foundation of China (Grant Nos. 60773078, 10571117) and the ShuGuang Plan of Shanghai Education Development Foundation (Grant No. 06SG42); M. A. Henning is supported in part by the South African National Research Foundation and the University of KwaZulu-Natal; this work is also supported by Shanghai Leading Academic Discipline Project (No. S30104)
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Shan, E.F., Sohn, M.Y., Yuan, X.D. et al. Domination number in graphs with minimum degree two. Acta. Math. Sin.-English Ser. 25, 1253–1268 (2009). https://doi.org/10.1007/s10114-009-7617-6
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DOI: https://doi.org/10.1007/s10114-009-7617-6