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 γω(G) and the irredundance number ir(G). We prove that γω(G) ≤ 5/2ir(G)-2 and this bound is sharp. Furthermore, for a tree T, we give a sufficient and necessary condition for γc(T) = γω(T) + k, where γc(G) is the connected domination number and 0 ≤ k ≤ γω(T) -1.