On the Complexity of Determining Whether there is a Unique Hamiltonian Cycle or Path

TL;DR

This paper investigates the computational complexity of determining the uniqueness of Hamiltonian cycles or paths in graphs, establishing that these problems are NP-hard and belong to the DP class, similar to the uniqueness problem in Boolean satisfiability.

Contribution

It proves that the problem of deciding the uniqueness of Hamiltonian cycles or paths has the same complexity as the U-SAT problem, extending known NP-completeness results.

Findings

Uniqueness problems are NP-hard.

They belong to the DP complexity class.

Complexity parallels the U-SAT problem.

Abstract

The decision problems of the existence of a Hamiltonian cycle or of a Hamiltonian path in a given graph, and of the existence of a truth assignment satisfying a given Boolean formula $C$, are well-known {\it NP}-complete problems. Here we study the problems of the {\it uniqueness} of a Hamiltonian cycle or path in an undirected, directed or oriented graph, and show that they have the same complexity, up to polynomials, as the problem U-SAT of the uniqueness of an assignment satisfying $C$. As a consequence, these Hamiltonian problems are {\it NP}-hard and belong to the class~{\it DP}, like U-SAT.