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

Abstract

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 ve 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], Ew) weakly induced by D, is connected, where Ew is the set of all edges having at least one vertex in D. The weakly connected domination number γw(G) of G is the minimum cardinality among all weakly connected dominating sets of G. In this article we characterize the graphs where γw(G) = τ (G). In particular, we focus our attention on bipartite graphs, regular graphs, unicyclic graphs, block graphs and corona graphs.