Hamiltonian orthogeodesic alternating paths

Let R be a set of red points and let B   be a set of blue points. The point set P=R∪BP=R∪B is called equitable   if ||B|−|R||⩽1||B|−|R||⩽1 and it is called general if no two points are vertically or horizontally aligned. An orthogeodesic alternating path on P is a path such that each edge is an orthogeodesic chain connecting points of different color and such that no two edges cross. We consider the problem of deciding whether a set of red and blue points admits a Hamiltonian   orthogeodesic alternating path, that is, an orthogeodesic alternating path visiting all points. We prove that every general equitable point set admits a Hamiltonian orthogeodesic alternating path and we present an O(nlog2n)-time algorithm for finding such a path, where n   is the number of points. On the other hand, we show that the problem is NPNP-complete if the path must be on the grid (i.e., vertices and bends have integer coordinates). Further, we show that we can approximate the maximum length of an orthogeodesic alternating path on the grid by a factor of 3, whereas we present a family of point sets with n   points that do not have a Hamiltonian orthogeodesic alternating path with more than n/2+2n/2+2 points. Additionally, we show that it is NPNP-complete to decide whether a given set of red and blue points on the grid admits an orthogeodesic perfect matching if horizontally aligned points are allowed. This contrasts a recent result by Kano (2009) [9] who showed that this is possible on every general point set.