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.