SCHWEITZER 1982 Problem 3

Let $G(V,E)$ be a connected graph, and let $d_G(x,y)$ denote the length of the shortest path joining $x$ and $y$ in $G$ . Let $r_G(x)= \max \{ d_G(x,y) : y \in V \ \}$ for $x \in V$ , and let $r(G)= \min \{ r_G(x) : x \in V\ \}$ . Show that if $r(G) \geq 2$ , then $G$ contains a path of length $2r(G)-2$ as an induced subgraph.

V. T. Sos