An efficient certifying algorithm for the Hamiltonian cycle problem on circular-arc graphs

Abstract

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is an evidence that can be used to authenticate the correctness of the answer. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to determine whether or not a graph contains a Hamiltonian cycle. The best result for the Hamiltonian cycle problem on circular-arc graphs is an O ( n 2 log n ) -time algorithm, where n is the number of vertices of the input graph. In fact, the O ( n 2 log n ) -time algorithm can be modified as a certifying algorithm although it was published before the term certifying algorithms appeared in the literature. However, whether there exists an algorithm whose time complexity is better than O ( n 2 log n ) for solving the Hamiltonian cycle problem on circular-arc graphs has been opened for two decades. In this paper, we present an O ( Δ n ) -time certifying algorithm to solve this problem, where Δ represents the maximum degree of the input graph. The certificates provided by our algorithm can be authenticated in O ( n ) time.