6.12 Speeding up the search for the absolute center In our discussion of the absolutewenter problem, it was pointed out that the quantity

provides a lower bound for the value of m(x) on a link (p, q) [cf. (6.26)]. This bound provides a very convenient test that facilitates the search for the absolute center of a graph.

Another bound for m(x) has been developed recently, as follows. For the link (p, q), let r be the farthest node from p (r N) and s the farthest node from q(s N) [i.e., m(p) = d(p, r) and m(q) = d(q, s)]. The quantities that will be used to develop the new bound are d(p, s) and d(q, r). Clearly, the quantity Max [d(x, r), d(x, s)] provides a lower bound on m(x).
 a. Prove the following result: The quantity Max{d(x, r), d(x, s)} may attain at most a single local minimum within (q, p), i.e., excluding the nodes p and q. If such a local minimum exists it is attained at a point x0 E (p, q) which is a distance away from node p along (p, q) and at which the value of Max{d(x, r), d(x, s)} attains the value L2(p, q) is then a lower bound for m(x) for all points x (p, q), excluding the nodes p and q. b. Show that L2(p, q) L1(p, q), that is, that L2(p, q) provides a better lower bound and thus a sharper test than L1(p, q) for speeding up the search for the absolute center of a graph.