For two nonadjacent vertices u

v in a graph G

using α(u

v) to denote the maximum cardinality of an independent vertex set of G containing both u and v

it is proved that if G is a 3 connected graph of order n and max{ d(u)

d(v)}≥(n+1)/2 for each pair of it is nonadjacent vertices u

v of G with\ 1≤| N(u)∩ N(v)|≤α(u

v)-1

then either G is Hamiltonian connected or else G belongs to a family of exceptional graphs.