Vertices Belonging to all or to no Minimum Vertex-Edge Dominating Sets in Trees

Abstract

Let G be a simple graph with vertex set V (G) and edge set E(G). A vertex v ∈ V (G) vertex-edge dominates every edge uv incident to v, as well as every edge adjacent to these incident edges. A set $D\subseteq V(G)$ is a vertex-edge dominating set if every edge of E(G) is vertex-edge dominated by a vertex of D. The vertex-edge dominating set of a minimum cardinality is called minimum vertex-edge dominating set. In this paper, we characterize the set of vertices that are in all or in no minimum vertex-edge dominating sets in trees.