In this paper, we give necessary and sufficient conditions for a subset S of the vertex set of a connected graph G to be a secure convex dominating set. Some realization problems will be given. In particular, we show that given positive integers k and n such that n ≥ 4 and 1 ≤ k ≤ n, there exists a connected graph G with |V (G)| = n and γ scon (G) = k. Also, we show that for any positive integers k, m and n such that 1 ≤ k ≤ m − 2 and m ≤ n, there exists a connected graph G with |V (G)| = n, γ scon (G) = m, and γ con (G) = k.

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] of vertices of a graph [Formula: see text] is an outer-convex dominating set if every vertex not in [Formula: see text] is adjacent to some vertex in [Formula: see text] and [Formula: see text] is a convex set. The outer-convex domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-convex dominating set of [Formula: see text]. An outer-convex dominating set of cardinality [Formula: see text] will be called a [Formula: see text]-[Formula: see text]. In this paper, we initiate the study and characterize the outer-convex dominating sets in the join of the two graphs.

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A dominating set S which is also convex is called a convex dominating set 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 γ rcon (G), is called the restrained convex domination number of G. In this paper we investigate the concept and give some important results.

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