On Hamiltonian alternating cycles and paths

TL;DR

This paper studies Hamiltonian alternating cycles and paths on bicolored point sets, relaxing planarity constraints to 1-plane, and provides algorithms and theoretical results for their existence and computation.

Contribution

It proves the existence of 1-plane Hamiltonian alternating cycles in general position and offers efficient algorithms for convex sets.

Findings

A 1-plane Hamiltonian alternating cycle always exists in general position.

Minimum crossing Hamiltonian cycles in convex position are 1-plane.

Algorithms run in O(n) and O(n^2) time for cycles and paths, respectively.

Abstract

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. Among them, we prove that a 1-plane Hamiltonian alternating cycle on a bicolored point set in general position can always be obtained, and that when the point set is in convex position, every Hamiltonian alternating cycle with minimum number of crossings is 1-plane. Further, for point sets in convex position, we provide $O(n)$ and $O(n^2)$ time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.