مشخصات پژوهش

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 2 S. A subset S of vertices of G is a geodetic set if I[S] = V. The geodetic number 1(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 K1 and pairwise disjoint complete graphs Kn1 ;Kn2 ; : : : ;Knr , that is, G = (Kn1 [ Kn2 [ : : : Knr ) + K1, where r  2, n1; n2; : : : ; nr are positive integers with n1 + n2 + : : : + nr = n 􀀀 1. In this paper we characterize all connected graphs G of order n  3 with 1(G) = n 􀀀 2.