Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $\left(K_r\cup Kₛ\cup Kₜ\right)\vee K₂$, 1 ≤ r ≤ s ≤ t or the graph $((n+1)/2)K₁\vee K_{(n-1)/2}$. It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph...