Question

# Let G = (V, E) be an undirected, unweighted graph with n = |V | vertices. The distance between two vertices u, v

∈ G is the length of the shortest path between them. A vertex cut of G is a subset S ⊆ V such that removing the vertices in S (as well as incident edges) disconnects G.

Show that if there exist u, v ∈ G of distance d > 1 from each other, that there exists a vertex cut of size at most (n − 2)/(d − 1). Assume G is connected.

I tried induction, but the inductive step seems to be a bit messy. What other proof approaches would work?

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