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Reconstruction of the Crossing Type of a Point Set from the Compatible Exchange Graph of Noncrossing Spanning Trees

  • Marcos Oropeza
  • Csaba D. TóthEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)

Abstract

Let P be a set of n points in the plane in general position. The order type of P specifies, for every ordered triple, a positive or negative orientation; and the x-type (a.k.a. crossing type) of P specifies, for every unordered 4-tuple, whether they are in convex position. Geometric algorithms on P typically rely on primitives involving the order type or x-type (i.e., triples or 4-tuples). In this paper, we show that the x-type of P can be reconstructed from the compatible exchange graph \(\mathcal {G}_1(P)\) of noncrossing spanning trees on P. This extends a recent result by Keller and Perles (2016), who proved that the x-type of P can be reconstructed from the exchange graph \(\mathcal {G}_0(P)\) of noncrossing spanning trees, where \(\mathcal {G}_1(P)\) is a subgraph of \(\mathcal {G}_0(P)\). A crucial ingredient of our proof is a structure theorem on the maximal sets of pairwise noncrossing edges (msnes) between two components of a planar straight-line graph on the point set P.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.California State University NorthridgeLos AngelesUSA
  2. 2.Tufts UniversityMedfordUSA

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