Carmelita M. Loquias, Enrico L. Enriquez
Let G be a connected simple graph. A convex dominating set S of V (G) is a secure convex dominating set of G if for each u ∈ V (G) \ S, there exists V ∈ S such that uv ∈ E(G) and the set (S \ {v }) ⋃ {u } is a convex dominating set of G. The minimum cardinality of a secure convex dominating set of G. denoted by γscon(G), is called the secure convex domination number of G. A convex dominating set S of V(G) is a restrained convex dominating set of G if for each u ∈ V (G) \ S, there exists z ∈ V(G) \ S such that uz ∈ E(G). The minimum cardinality of a restrained convex dominating set of G, denoted by γrcon(G), is called the restrained convex domination number of G. In this paper, we show that every integers a, b, and n with 1 ≤ a ≤ b ≤ n(a ≠ n - 1) is realizable as restrained convex domination number, secure convex domination number and order of a graph G respectively. Further, we give the secure convex domination number and restrained convex domination number of some special graphs and give some important results. © Research India Publications.
Mathematics Department, School of Arts and Sciences, University of San Carlos, Philippines
