Let G = (V,E) be a connected graph and a  V be an articulation point of G. How can be proved that there is an edge e  G incident with a with the property that e doesn't belong to a perfect matching of G?Let's say that there are two vertices u,v  G so that the only paths between them go through a. Then there is no edge directly connecting u and v. It means that a has at least 2 vertices(which is pretty obvious) and knowing that the property of a perfect matching is that the degree of any vertex is equal to 1, it means that there is at least one edge incident to a that doesn't belong to the perfect matching. But I'm not sure, is it okay?
Edge incident to cut-vertex not in perfect matching of a graph
graphs;graph theory;matching