Insertible vertices,neighborhood intersections,and hamiltonicity

Let G be a simple undirected graph of order n. For an independent set S ? V(G) of k vertices, we define the k neighborhood intersections S_i = {v ? V(G)\S|N(v) ∩ S| = i}, 1 ≦ i ≦ k, with s_i = |S_i|. Using the concept of insertible vertices and the concept of neighborhood intersections, we prove the following theorem.