On weakly connected domination in graphs II

A dominating set D is a weakly connected dominating set of a connected graph G=(V,E) if (V,E∩(D×V)) is connected. The weakly connected domination number of G, denoted γwc(G), is min{|S||S is a weakly connected dominating set of G}. We characterize graphs G for which γ(H)=γwc(H) for every connected induced subgraph H of G, where γ is the domination number of a graph. We provide a constructive characterization of trees T for which γ(T)=γwc(T). Lastly, we constructively characterize the trees T in which every vertex belongs to some weakly connected dominating set of cardinality γwc(T).