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

A set SS of vertices in a graph GG is a dominating set of GG if every vertex of V(G)∖SV(G)∖S is adjacent to some vertex in SS. The minimum cardinality of a dominating set of GG is the domination number of GG, denoted as γ(G)γ(G). Let PnPn and CnCn denote a path and a cycle, respectively, on nn vertices. Let k1(F)k1(F) and k2(F)k2(F) denote the number of components of a graph FF that are isomorphic to a graph in the family {P3,P4,P5,C5}{P3,P4,P5,C5} and {P1,P2}{P1,P2}, respectively. Let LL be the set of vertices of GG of degree more than 2, and let G−LG−L be the graph obtained from GG by deleting the vertices in LL and all edges incident with LL. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749–762] showed that if GG is a connected graph of order n≥8n≥8 with δ(G)≥2δ(G)≥2, then γ(G)≤2n/5γ(G)≤2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277–295] showed that if GG is a graph of order nn with δ(G)≥3δ(G)≥3, then γ(G)≤3n/8γ(G)≤3n/8. As an application of Reed’s result, we show that if GG is a graph of order n≥14n≥14 with δ(G)≥2δ(G)≥2, then γ(G)≤38n+18k1(G−L)+14k2(G−L).