Outer-convex domination in the corona of graphs

Abstract

Let G be a connected simple graph. A subset S of a vertex set V (G) is called an outer-convex dominating set of G if for every vertex v ∈ V (G)\S, there exists a vertex x ∈ S such that xv is an edge of G and V (G)\S is a convex set. The outer-convex domination number of G, denoted by γecon(G), is the minimum cardinality of an outerconvex dominating set of G. In this paper, we show that every integers a, b, c, and n with a ≤ b ≤ c ≤ n − 1 is realizable as domination number, outer-connected domination number, outer-convex domination number, and order of G respectively. Further, we give the characterization of the outer-convex dominating set in the corona of two graphs and give its corresponding outer-convex domination number.