Any undirected and simple graph G = (V, E), where V and E denote the vertex set and the edge set of G, is called Hamiltonian if it contains a cycle that visits each vertex of G exactly once. Ore proved that G is Hamiltonian if degG(u) + degG(v) >= n holds for every nonadjacent pair of vertices u and v in V, where n is the total number of distinct vertices of G. Su, Shih, and Kao proved that any graph G satisfying Ore’s condition remains Hamiltonian after remov-ing any one vertex x € V unless G belongs to one of two exceptional families of graphs. This paper proves that G − {x, y} is Hamiltonian for any two vertices x, y € V, unless G belongs to one of the eight exceptional families of graphs, denoted by i, where i € {1, …, 8}.