Outer-convex domination in graphs

Let G be a connected simple graph. A set S of vertices of a graph G is an outer-convex dominating set if every vertex not in S is adjacent to some vertex in S and V (G)\ S is a convex set. The outer-convex domination number of G, denoted by (gamma) over tilde (con)(G), is the minimum cardinality of an outer-convex dominating set of G. An outer-convex dominating set of cardinality (gamma) over tilde (con)(G) will be called a (gamma) over tilde (con)-set. In this paper, we initiate the study and characterize the outer-convex dominating sets in the join of the two graphs.