Weakly Connected Domination in Graphs Resulting from Binary Operations

Abstract

Let G = ( V (G),E (G) ) be a connected undirected graph. The closed neighborhood of any vertex v \(\epsilon\) V (G) is N_G[v] = {u \(\epsilon\) V (G) : uv \(\epsilon\) E (G) } U {v}. For \(\subseteq\) V  (G) , the closed neighborhood of C is N[C] = \(U_{(vec)}\) N_G [v]. A dominating set C \(\subseteq\) V (G) is a weakly connected dominating set in G if the subgraph \(\langle\)C\(\rangle\)_w = (N_G[C],E_W) weakly induced by C is connected, where E_W is the set of all edges with at least one vertex in C. The weakly connected domination number \(\gamma\)_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 K_r-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.