3. Let G be a k-connected graph, and let it be a new graph built from G by adding a new vertex v and making it adjacent

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3 Let G Be A K Connected Graph And Let It Be A New Graph Built From G By Adding A New Vertex V And Making It Adjacent 1 (18.72 KiB) Viewed 24 times

3. Let G be a k-connected graph, and let it be a new graph built from G by adding a new vertex v and making it adjacent to k of the vertices of G. (a) Reasoning directly from the definition, prove that H is also k-connected. (b) If s is a vertex of G and T is a set of vertices, then an s-T fan is a collection of 8 - paths for eacht € T that share no vertices other than s. Use Menger's theorem and the previous part of this problem to prove the following: Theorem 3.1. If G is a k-connected graph, is a verter of G, and TCV(G) with NT and T|=k, then G contains ans-T fan.