Similarities and Differences Between the Vertex Cover Number and the Weakly Connected Domination Number of a Graph

A vertex cover of a graph G = (V, E) is a set X ⊂ V such that each edge of G is incident to at least one vertex of X. The vertex cover number τ(G) is the minimum cardinality of a vertex cover of G. A dominating set D ⊆ V is a weakly connected dominating set of G if the subgraph G[D] _w = (N[D], E _w ) weakly induced by D, is connected, where E_w is the set of all edges having at least one vertex in D. The weakly connected domination number γ_w(G) of G is the minimum cardinality among all weakly connected dominating sets of G. In this article we characterize the graphs where γ_w(G) = τ(G). In particular, we focus our attention on bipartite graphs, regular graphs, unicyclic graphs, block graphs and corona graphs.