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

TL;DR

This paper introduces a new, more general graph closure concept based on neighborhood-equivalence that preserves Hamiltonicity, extending previous methods and enabling denser closures for better graph analysis.

Contribution

It generalizes existing closure concepts by broadening neighborhood-equivalence criteria, resulting in a denser graph closure that still preserves Hamiltonicity.

Findings

The new closure preserves Hamiltonicity in broader classes of graphs.

Denser closures are achievable without losing Hamiltonicity properties.

The approach extends and unifies previous closure methods.

Abstract

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.