We enhance classical conditions for Hamiltonian cycles in a graph G by adding or deleting certain edges that cannot lie in such cycles. The necessary condition of 1-toughness is that for all \(S\subseteq V(G)\) , the graph \(G-S\) has at most \(\left| S \right| \) components. The sufficient condition by Ore is that the degrees of any two nonadjacent vertices sum to at least \(\left| V(G) \right| \) . The reduction operation involving deletions enables proving the absence of Hamiltonian cycles in some graphs that are 1-tough. The operation that adds edges enables guaranteeing Hamiltonian cycles in some graphs that do not satisfy Ore’s condition.
