Weakly Connected Domination in Graphs Resulting from Binary Operations

Let G = ( V (G),E (G) ) be a connected undirected graph. The closed neighborhood of any vertex v V (G) is NG[v] = {u V (G) : uv E (G) } U {v}. For V (G) , the closed neighborhood of C is N[C] = NG [v]. A dominating set C V (G) is a weakly connected dominating set in G if the subgraph C w = (NG[C],EW) weakly induced by C is connected, where EW is the set of all edges with at least one vertex in C. The weakly connected domination number w (G) of G is the minimum cardinality among all weakly connected dominating sets in G. In this paper, we characterized the weakly connected dominating sets in the Kr-gluing of complete graphs and corona of graphs. As con- sequences, we determined the weakly connected domination number of the aforementioned graphs. Weakly connected domination number in the join of graphs is also determined.