Given a graph G=(V, E), a vertex is said to ve-dominate an edge if it is either incident with the edge or adjacent to one of its endpoints. A set of vertices is a ve-dominating set if it ve-dominates every edge of the graph. We introduce the class of well-ve-dominated graphs, defined as graphs in which all minimal ve-dominating sets have the same cardinality. After establishing several general structural properties of well-ve-dominated graphs, we show that recognizing whether a graph belongs to this class is co--NP--complete, highlighting the computational difficulty of the problem. Our main result is a complete structural characterization of well-ve-dominated trees, which yields a simple linear-time recognition algorithm and a constructive description of all trees in this class.

翻译：给定图 G=(V, E)，若一个顶点与某条边相关联或与该边某一端点相邻，则称该顶点VE-支配该边。一个顶点集合若VE-支配图中所有边，则称为VE-支配集。本文引入良VE-支配图类，定义为所有极小VE-支配集具有相同基数的图。在建立良VE-支配图的若干一般结构性质后，我们证明判定一个图是否属于该类是co-NP-完全的，这揭示了该问题的计算复杂性。我们的主要成果是对良VE-支配树给出了完整结构刻画，由此导出一个简单的线性时间判定算法，并构造性地描述了该类中所有树的结构。