Let G(V,E) be a connected graph, and let dG(x,y) denote the length of the shortest path joining x and y in G. Let r_G(x)\equal{} \max \{ d_G(x,y) : \; y \in V \ \} for x∈V, and let r(G)\equal{} \min \{ r_G(x) : \;x \in V\ \}. Show that if r(G)≥2, then G contains a path of length 2r(G)\minus{}2 as an induced subgraph.
V. T. Sos combinatorics proposedcombinatorics