It is well known that in a bipartite (and more generally in a König-Egerváry) graph, the size of the minimum vertex cover is equal to the size of the maximum matching. We first address the question whether (and if not, when) this property still holds in a König-Egerváry graph if we consider vertex covers containing a given subset of vertices. We characterize such graphs using the classic notions of alternating paths and flowers used in Edmonds' matching algorithm. We then use the notions of alternating paths and flowers in König-Egerváry graphs to give a complete characterization of such graphs that have a unique minimum vertex cover.

众所周知，在二部图（更常见的是在König-Egerváry中）中，最小顶点覆盖的大小等于最大匹配的大小。我们首先要解决的问题是，如果考虑包含给定顶点子集的顶点覆盖，则该属性是否（以及是否在何时）仍保留在König-Egerváry图中。我们使用Edmonds匹配算法中使用的交替路径和花朵的经典概念来表征此类图。然后，我们使用König-Egerváry图中的交替路径和花朵的概念来完全表征具有唯一最小顶点覆盖的图。