Doubly connected domination in the join and Cartesian product of some graphs
Benjier H. Arriola, Sergio R. Canoy · Asian-European Journal of Mathematics · 2014
Let G be a simple connected graph. A connected dominating set S ⊂ V(G) is called a doubly connected dominating set of G if the subgraph 〈V(G)\S〉 induced by V(G)\S is connected. We show that given any three positive integers a, b, and c with 4 ≤ a ≤ b ≤ c, where b ≤ 2a, there exists a connected graph G such that a = γr(G), b = γtr(G), and c = γcc(G), where γr, γtr, and γcc are, respectively, the restrained domination, total restrained domination, and doubly connected domination parameters. Also, we characterize the doubly connected dominating sets in the join of any graphs and Cartesian product of some graphs. The corresponding doubly connected domination numbers of these graphs are also determined.