Let G be a connected graph of order n and
X={x∈V:d(x)≥n/2}. Suppose
|X|≥3 and G satisfies the
modified Fan's condition. We show that the vertices of the block
B of G containing X form a cycle. This generalizes a result
of Fan. We also give an efficient algorithm to obtain such a
cycle. The complexity of this algorithm is O(n2). In case G is 2-connected, the condition |X|≥3 can be removed and G is hamiltonian.