Minimal TSP Tour is coNP-Complete

Abstract

The problem of deciding if a Traveling Salesman Problem (TSP) tour is minimal was proved to be coNP-complete by Papadimitriou and Steiglitz. We give an alternative proof based on a polynomial time reduction from 3SAT. Like the original proof, our reduction also shows that given a graph $G$ and an Hamiltonian path of $G$, it is NP-complete to check if $G$ contains an Hamiltonian cycle (Restricted Hamiltonian Cycle problem).