Doubly connected domination in the join and Cartesian product of some graphs

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.

-Tuple Total Restrained Domination in Complementary Prisms

In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.