Compatible Geometric Matchings

Abstract

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M of the same set of n points,such that each M i is compatible with M i+1 . This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.

Compatible spanning trees

“…, T k−1 , T k = T b , with k ∈ O(log n), in which every two consecutive trees are compatible [2]. Similar results have been obtained for other configurations, such as matchings [3,15], and (in the negative) for pointed pseudotriangulations [4]. A major open problem in the area is whether this kind of sequences exist or not for polygonizations, or for spanning paths, of any given point set.…”

“…As this is not always possible, several alternative conjectures and partial results arose [21,28,23], culminating with the proof by Hoffmann and Tóth [14] that a compatible Hamiltonian cycle always exists, i.e., a polygonization P of S in which all the segments in M are either sides of the polygon P , or internal diagonals of P , or external diagonals of P . A quite similar problem was posed and discussed in [3]: Given a non-crossing perfect matching on a point set S, one wants to find another non-crossing perfect matching that is compatible and edge-disjoint with the given one. A solution in the affirmative has been obtained very recently by Ishaque et al [19] when there is an even number of segments (disjointness is not always possible when the number is odd, as proved in [3]).…”

“…A quite similar problem was posed and discussed in [3]: Given a non-crossing perfect matching on a point set S, one wants to find another non-crossing perfect matching that is compatible and edge-disjoint with the given one. A solution in the affirmative has been obtained very recently by Ishaque et al [19] when there is an even number of segments (disjointness is not always possible when the number is odd, as proved in [3]). …”

Matching Points with Things

“…Note that p i can be in at most one unit. If it is, we determine if ∆ i reduces this unit by case (2). Likewise, for every point p j in A, we locate p j in B to determine if it is inside a unit in B and apply the appropriate reductions.…”

“…The non-crossing requirement in our problems is quite natural in geometric scenarios (see for example [25,2,3]), and the family of geometric problems that we consider has several applications; these applications include geometric shape matching [4,13,17,18], colour-based image retrieval [13], music score matching [26], and computational biology [14,16].…”