Vertex-edge domination

Abstract

Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertexedge 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 γ_ve(G). We present bounds on γ_ve(G) and relationships between γ_ve(G) and other domination related parameters. Since any ordinary dominating set is also a vertex-edge dominating set, it follows that γ_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.