Outer-Connected Inverse Domination in Graphs

Let be a connected simple graph. A subset of is a dominating set of if for every , there exists such that. A set is said to be an outer-connected dominating set in if is dominating and either or is connected. Let be a minimum dominating set of . A nonempty subset is an outer-connected inverse dominating set of , if is an inverse dominating set with respect to and the subgraph induced by is connected. The outer-connected inverse domination number of , is denoted by , that is, the minimum cardinality of an outer-connected inverse dominating set of . In this paper, we initiate the study of the concept and give the outer-connected inverse domination number of some special graphs. Further, we give the characterization of the outer-connected inverse dominating set in the join of two nontrivial connected graphs.