Total Vertex-Edge Domination Number of Some Graphs

Abstract

For a graph \(G=(V,E)\), a vertex \(u\) in \(G\) vertex-edge dominates (or simply \(ve\)-dominates) an edge \(e\) in \(G\) if \(u\) is incident to \(e\), or \(u\) is incident to an edge that is adjacent to \(e\). A set \(S \subseteq V\) is a vertex-edge dominating set if for all edges \(e \in E\), there exists a vertex \(v \in S\) that \(ve\)-dominates \(e\). The minimum cardinality of a \(ve\)-dominating set of graph \(G\) is called the vertex-edge domination number, and is denoted by \(\gamma_{ve} (G)\). A \(ve\)-dominating set is said to be total if its induced subgraph has no isolated vertices, that is, every vertex of \(S\) has a neighbor in \(S\). The total vertex-edge domination number, denoted by \(\gamma_{ve}^t (G)\), of \(G\) is the minimum cardinality of a total \(ve\)-dominating set. This study focuses on the total vertex-edge domination number in graphs. The authors determined its exact value for the cycle graph \(C_n\), as well as for the join and corona of two arbitrary graphs \(G\) and \(H\). A complete characterization of graphs with total vertex-edge domination number equal to two was also established. Furthermore, a general bound for the total vertex-edge domination number of the Cartesian product \(G\times K_n\), where \(n\geq 4\), was derived.