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.