Influence of the Edge Removal, Edge Addition and Edge Subdivision on the Double Vertex–Edge Domination Number of a Graph

Abstract

A vertex v of a graph $G=(V,E)$ is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set $S \subseteq V$ is a vertex–edge dominating set (double vertex–edge dominating set, respectively) if every edge of E is ve-dominated by at least one vertex (at least two vertices) of S. The minimum cardinality of a vertex–edge dominating set (double vertex–edge dominating set, respectively) of G is the vertex–edge domination number $\gamma _{ve}(G)$ (the double vertex–edge domination number $\gamma _{dve}(G)$, respectively). The influence of edge removal, edge addition and edge subdivision on the double vertex–edge domination number of a graph are investigated in this paper.