Inverse Doubly Connected Domination in the Lexicographic Product of Two Graphs

Let G be a nontrivial connected graph. A dominating set D⊆V (G) is called a doubly connected dominating set of G if both 〈D〉 and 〈V (G) 〉 are connected. Let D be a minimum connected dominating set of G. If S⊆V (G) is a connected dominating set of G, then S is called an inverse doubly connected dominating set of G with respect to D. Furthermore, the inverse doubly connected domination number, denoted by γcc^ (-1) (G) is the minimum cardinality of an inverse doubly connected dominating set of G. An inverse doubly connected dominating set of cardinalities γcc^ (-1) (G) is called γcc^ (-1) -set. In this paper, we characterized the inverse doubly connected domination in the lexicographic product of two graphs and give some important results.