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| In the article one advanced some important concepts: cut-off-able edge, cut-off-unable edge, openable edge, unopenable edge, openable region, unopenable region, closed corridor, opened corridor, adjoint subgraph of the vertex of $d(v) geq 3$, the subgraph of double bridges with double chains, the forbidden subgraph to the Hamiltonian, etc. And one advanced and shown the necessary and sufficient condition of deciding an arbitrary undirected graph except the one of knight\\\\\\\\\\\\\\\ |
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| Keywords:Hamiltonian, Hamiltonian cycle, Hamiltonian graph, vertex, edge, forbidden subgraph, corridor, computability, complexity. |
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