Domination number in graphs with minimum degree two

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.