On the p-domination number of cactus graphs

Let p be a positive integer and G = (V, E) a graph. A subset S of V is a p-dominating set if every vertex of V − S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number γp(G). It is proved for a cactus graph G that γp(G) 6 (|V |+ |Lp(G)|+c(G))/2, for every positive integer p > 2, where Lp(G) is the set of vertices of G of degree at most p − 1 and c(G) is the number of odd cycles in G.