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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 , 1 ≤ r ≤ s ≤ t or the graph .
It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph...
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