Convex 2-Domination in Graphs.

Let G be a connected graph. A set S ⊆ V (G) is convex 2-dominating if S is both convex and 2-dominating. The minimum cardinality among all convex 2-dominating sets in G, denoted by γ2con(G), is called the convex 2-domination number of G. In this paper, we initiate the study of convex 2- domination in graphs. We show that any two positive integers a and b with 6 ≤ a ≤ b are, respectively, realizable as the convex domination number and convex 2-domination number of some connected graph. Furthermore, we characterize the convex 2-dominating sets in the join, corona, lexicographic product, and Cartesian product of two graphs and determine the corresponding convex 2-domination number of each of these graphs. [ABSTRACT FROM AUTHOR]