Vertex covers and secure domination in graphs

Let G = (V, E) be a graph and let S subset of V. The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a secure dominating set of G if for each u is an element of V \ S, there exists a vertex v is an element of S such that uv is an element of E and (S \ {v}) boolean OR {u} is a dominating set of G. The minimum cardinality of a secure dominating set in G is the secure domination number gamma(s)(G) of G. We show that if G is a connected graph of order n with minimum degree at least two that is not a 5-cycle, then gamma(s)(G) <= n/2 and this bound is sharp. Our proof uses a covering of a subset of V (G) by vertex-disjoint copies of subgraphs each of which is isomorphic to K-2 or to an odd cycle.