Vertex-edge domination is a natural variant of domination in which a vertex ve-dominates an edge whenever it is incident to the edge or adjacent to one of its endpoints. A set of vertices is a vedominating set if it ve-dominates every edge of the graph. In this paper, we introduce the notion of wellve- dominated graphs, namely graphs in which all minimal ve-dominating sets have the same cardinality. Equivalently, these are graphs whose minimal isolating sets all have the same cardinality; we refer to them as well-isolated graphs. We prove that recognizing well-ve-dominated graphs is co-NP-complete. We determine perfectly well-ve-dominated graphs, the largest hereditary subclass, and obtain both a complete forbidden induced subgraph characterization and a linear-time recognition algorithm. In particular, every connected nontrivial member of this subclass has vertex-edge domination and isolation number one. Finally, we characterize the class of well-ve-dominated trees. More precisely, we prove that well-ve-domination and well-ve-coveredness coincide on nontrivial trees, where well-ve-coveredness requires all minimal independent ve-dominating sets to have the same cardinality. The characterization yields a linear-time recognition algorithm and identifies the reduced members with the extremal trees attaining equality in the sharp isolation bound.

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