Abstract
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.
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