A note on weakly connected domination number in graphs

Let G be a connected graph. A weakly connected dominating set of G is a dominating set D such that the edges not incident to any vertex in D do not separate the graph G. In this paper, we first consider the relationship between weakly connected domination number gamma(w)(G) and the irredun-dance number ir(G). We prove that gamma(w)(G) <= 5/2 ir(G)-2 and this bound is sharp. Furthermore, for a tree T, we give a sufficient and necessary condition for gamma(c)(T) = gamma(w)(T) + k, where N(G) is the connected domination number and 0 <= k <= gamma(w).,(T) - 1.