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Proof of a Characterization of Cut Vertices | Graph Theory
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A vertex v, of a connected graph G, is a cut vertex if and only if there exist two vertices u and w - distinct from v - such that every u-w path in G contains v. We'll be proving this characterization of cut vertices in today's video graph theory lesson!
The second direction is to assume that v is a vertex of G that lies on every u-w path for two vertices u and w, neither equal to v in G. Then we show v must be a cut vertex of G by analyzing the graph G-v, which clearly cannot have a u-w path. Thus G-v must be disconnected, so v is a cut vertex.
I hope you find this video helpful, and be sure to ask any questions down in the comments!
+WRATH OF MATH+
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The second direction is to assume that v is a vertex of G that lies on every u-w path for two vertices u and w, neither equal to v in G. Then we show v must be a cut vertex of G by analyzing the graph G-v, which clearly cannot have a u-w path. Thus G-v must be disconnected, so v is a cut vertex.
I hope you find this video helpful, and be sure to ask any questions down in the comments!
+WRATH OF MATH+
Follow Wrath of Math on...
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