An upper bound on the domination number of a graph with minimum degree 2

A set S of vertices in a graph G is a dominating set of G if every vertex of V (G)\S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ (G). Let Pn and Cn denote a path and a cycle, respectively, on n vertices. Let k1(F) and k2(F) denote the number of components of a graph F that are isomorphic to a graph in the family {P3, P4, P5,C5} and {P1, P2}, respectively. Let L be the set of vertices of G of degree more than 2, and let G − L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749–762] showed that if G is a connected graph of order n ≥ 8 with δ(G) ≥ 2, then γ (G) ≤ 2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277–295] showed that if G is a graph of order n with δ(G) ≥ 3, then γ (G) ≤ 3n/8. As an application of Reed’s result, we show that if G is a graph of order n ≥ 14 with δ(G) ≥ 2, then γ (G) ≤ 3 8 n + 1 8 k1(G − L)+ 1 4 k2(G − L). c © 2008 Elsevier B.V. All rights reserved.