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 P n and C n denote a path and a cycle, respectively, on n vertices. Let k 1 ( F ) and k 2 ( F ) denote the number of components of a graph F that are isomorphic to a graph in the family { P 3 , P 4 , P 5 , C 5 } and { P 1 , P 2 } , 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 ) ≤ 3 8 n + 1 8 k 1 ( G − L ) + 1 4 k 2 ( G − L ) .