An O(n² log n) Algorithm for the Hamiltonian Cycle Problem on Circular-Arc Graphs

A Hamiltonian cycle in a graph G is a simple cycle in which each vertex of G appears exactly once. The Hamiltonian cycle problem involves testing whether a Hamiltonian cycle exists in a graph, and finds one if such a cycle does exist. It is well known that the Hamiltonian cycle problem is one of the classic NP-complete problems on general graphs. Shih et al. solved the Hamiltonian cycle problem on circular-arc graphs in O(n log n) time [36], where n is the number of vertices of the input graph. Whether there exists a more efficient algorithm for solving the Hamiltonian cycle problem on circular-arc graphs has been opened for a decade. In this paper, we present an O(∆n)-time algorithm to solve it, where ∆ denotes the maximum degree of the input graph.