Convex 2-Domination in Graphs

Abstract

Let $G$ be a connected graph. A set $S \subseteq V(G)$ is \textit{convex $2$-dominating} if $S$ is both convex and $2$-dominating. The minimum cardinality among all convex $2$-dominating sets in $G$, denoted by $\gamma_{2con}(G)$, is called the \textit{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 $6 \le a \le b$ 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.