Well ve-covered graphs

A vertex u of a graph G = (V, E) ve-dominates every edge incident to u as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved-set for short) if every edge of E is ve-dominated by at least one vertex in S. A ved-set is independent if its vertices are pairwise non-adjacent. The independent ve-domination number ive(G) is the minimum cardinality of an independent ved-set and the upper independent ve-domination number βve(G) is the maximum cardinality of a minimal independent ved-set of G. In this paper, we are interesting in graphs G such that ive(G) = βve(G), which we call well ve-covered graphs. We show that recognizing well ve-covered graphs is co-NP-complete, and we present a constructive characterization of well ve-covered trees.