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) {set minus} 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 P_n and C_n 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) ≤ 2 n / 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) ≤ 3 n / 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) ≤ frac(3, 8) n + frac(1, 8) k₁ (G - L) + frac(1, 4) k₂ (G - L).