Order on Order Types

Given P and P′, equally sized planar point sets in general position, we call a bijection from P to P′ crossing-preserving if crossings of connecting segments in P are preserved in P′ (extra crossings may occur in P′). If such a mapping exists, we say that P′ crossing-dominates P, and if such a mapping exists in both directions, P and P′ are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.