A linear algorithm for finding Hamiltonian cycles in 4-connected maximal planar graphs

The Hamiltonian cycle problem is one of the most popular NP-complete problems, and remains NP-complete even if we restrict ourselves to a class of (3-connected cubic) planar graphs [5,9]. Therefore, there seems to be no polynomial-time algorithm for the Hamiltonian cycle problem. However, for certain (nontrivial) classes of restricted graphs, there exist polynomial-time algorithms [3,4,6]. In fact, employing the proof technique used by Tutte [lo], Gouyou-Beauchamps has given an 0(n3) time algorithm for finding a Hamiltonian cycle in a 4-connected planar graph G, where n is the number of vertices of G [6]. Although such a graph G always has a Hamiltonian cycle [lo], it is not an easy matter to actually find a Hamiltonian cycle of G. However, for a little more restricted class of graphs, i.e., the class of 4-connected maximal planar graphs, we can construct an efficient algorithm. One can easily design an O(n*) time algorithm to find a Hamiltonian cycle in a 4-connetted maximal planar graph G with n vertices, entirely based on Whitney’s proof of his theorem [l 11. In this paper, we present an efficient algorithm for the problem, based on our simplified version of Whitney’s proof of his result. We employ ‘divide and conquer’ and some other techniques in the algorithm. The computational complexity of our algorithm is linear, thus optimal to within a constant factor.