Let G = (V (G), E(G)) be a connected simple graph. A connected dominating set S of V (G) is a secure connected 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 connected dominating set of G. The minimum cardinality of a secure connected dominating set of G, denoted by γ sc (G), is called the secure connected domination number of G. We characterized secure connected dominating set in terms of the concept of external private neighborhood of a vertex. Also, we give necessary and sufficient conditions for connected graphs to have secure connected domination number equal to 1 or 2. The secure connected domination numbers of graphs resulting from some binary operations are also investigated.

In this paper we revisit the concept of secure connected domination in graphs. In particular, we characterized secure connnected dominating sets in the join and composition of graphs and obtained the corresponding upper bounds or exact values of the secure connected domination numbers of these graphs. A rectification of a result obtained in [1] is given.

Let G = (V (G), E(G)) be a connected simple graph. A connected dominating set S of V (G) is a restrained connected dominating set of G if for each u ∈ V (G) \ S, there exists v ∈ V (G) \ S such that uv ∈ E(G). The minimum cardinality of a restrained connected dominating set of G, denoted by γ rc (G), is called the restrained connected domination number of G. We revisit these concepts for some graphs and give some important results.