A generalized closure concept based on neighborhood-equivalence and preserving graph Hamiltonicity

A graph is Hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is Hamiltonian is known as a NP-complete problem and no satisfactory characterization for these graphs has been found. In 1976 Bondy and Chvatal introduced a way to get round the Hamiltonicity problem complexity by using a closure of the graph. This closure is a supergraph of G which preserves Hamiltonicity, that is, which is Hamiltonian if and only if G is. Since this seminal work, several closure concepts preserving Hamiltonicity were introduced. In particular Ryjacek defined in 1997 a closure concept for claw-free graphs based on local completion. The completion is performed for every eligible vertex of the graph. Extending these works, Vallee and Bretto recently introduced a new closure concept preserving Hamiltonicity and based on local completion. The local completion is performed for each neighborhood-equivalence eligible vertex of the graph. In this article, we generalize the main results of Vallee and Bretto by introducing a broader notion of neighborhood equivalence eligibility, allowing the definition of a denser graph closure which still preserves Hamiltonicity.