We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n 2 ) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.
Claverol et al. REF obtain the same result, but requiring the geometric Hamiltonian alternating cycle to be 1-plane (that is, every edge is crossed at most once).
On Hamiltonian alternating cycles and paths
ArXiv
ArXiv
Mathematics Computer Science