Vertex-Edge Domination

Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertex-edge domination where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. The minimum cardinality of a vertex-edge dominating set of a graph G is the vertex-edge domination number gamma(ve)(G). We present bounds on gamma(ve)(G) and relationships between gamma(ve)(G) and other domination related parameters. Since any ordinary dominating set is also a vertex-edge dominating set, it follows that gamma(ve)(G) is bounded above by the domination number of G. Our main result characterizes the trees having equal domination and vertex-edge domination numbers.