Graphs with Large Geodetic Number

For two vertices u and v of a graph G, the set I[u, v] consists of all vertices lying on some u - v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v is an element of S. A subset S of vertices of G is a geodetic set if I[S] = V. The geodetic number g(G) is the minimum cardinality of a geodetic set of G. It was shown that a connected graph G of order n >= 3 has geodetic number n - 1 if and only if G is the join of K-1 and pairwise disjoint complete graphs K-n1, K-n2,..., K-nr, that is, G = (K-n1 boolean OR K-n2 boolean OR ... K-nr) + K-1, where r >= 2, n(1), n(2), ...., n(r) are positive integers with n(1) + n(2) + ... + n(r) = n - 1. In this paper we characterize all connected graphs G of order n >= 3 with g(G) = n - 2.