Weakly Convex and Convex Domination Numbers of Some Products of Graphs

Abstract

If $G=(V,E)$ is a simple connected graph and $a,b\in V$, then a shortest $(a–b)$ path is called an $(a–b)$-geodesic. A set $X\subseteq V$ is called weakly convex in $G$ if for every two vertices $a,b\in X$ there exists an $(a–b)$-geodesic whose all vertices belong to $X$. A set $X$ is convex in $G$ if for every $a,b\in X$ all vertices from every $(a–b)$-geodesic belong to $X$. The weakly convex domination number of a graph $G$ is the minimum cardinality of a weakly convex dominating set in $G$, while the convex domination number of a graph $G$ is the minimum cardinality of a convex dominating set in $G$. In this paper, we consider weakly convex and convex domination numbers of Cartesian products, joins, and coronas of some classes of graphs.