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.