Bounds relating the weakly connected domination number to the total domination number and the matching number

Let G=(V,E)G=(V,E) be a connected graph. A dominating set SS of GG is a weakly connected dominating set of GG if the subgraph (V,E∩(S×V))(V,E∩(S×V)) of GG with vertex set VV that consists of all edges of GG incident with at least one vertex of SS is connected. The minimum cardinality of a weakly connected dominating set of GG is the weakly connected domination number, denoted γwc(G). A set SS of vertices in GG is a total dominating set of GG if every vertex of GG is adjacent to some vertex in SS. The minimum cardinality of a total dominating set of GG is the total domination number γt(G)γt(G) of GG. In this paper, we show that 12(γt(G)+1)≤γwc(G)≤32γt(G)−1. Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of GG is called the matching number of GG, denoted α′(G)α′(G). We also establish that γwc(G)≤α′(G), and show that γwc(T)=α′(T) for every tree TT.