On weakly connected domination in graphs

Let G = (V,E) be a connected undirected graph. For any vertex v ∈ V , the closed neighborhood of v is N [v] = {v} ∪ {u ∈ V | uv ∈ E }. For S ⊆ V , the closed neighborhood of S is N [S] = ⋃ v∈S N [v]. The subgraph weakly induced by S is 〈S〉w = (N [S], E ∩ (S × N [S])). A set S is a weakly-connected dominating set of G if S is dominating and 〈S〉w is connected. The weakly-connected domination number is γw(G) = min{ |S| | S is a weakly-connected dominating set of G }, and the upper weakly-connected domination number is Γw(G) = max{ |S| | S is a minimal weakly-connected dominating set of G }. The weakly-connected independent domination number, iw(G) = min{ |S| | S is a weakly-connected independent dominating set of G }, and the upper weakly-connected independent domination number, βw(G) = max{ |S| | S is a weakly-connected independent dominating set of G }, always exist. We show that the problems of computing γw, iw, βw, and Γw are NP-hard; show that there exist linear algorithms to compute γw(T ) and Γw(T ) for a tree T and iw(G) and βw(G) for a connected bipartite graph G; and provide sharp lower and upper bounds on γw(G) for a connected graph G.