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.