Covering the Edges of a Complete Geometric Graph with Convex Polygons

Rom Pinchasi, Oren Yerushalmi

Research output: Contribution to journalArticlepeer-review

Abstract

Given a set P of m⩾3 points in general position in the plane, we want to find the smallest possible number of convex polygons with vertices in P such that the edges of all these polygons contain all the m2 straight line segments determined by the points of P. We show that if m is odd, the answer is (m2-1)/8 regardless of the choice of P. In this case there is even a partitioning of the edges of the complete geometric graph on m vertices into (m2-1)/8 convex polygons. The answer in the case where m is even depends on the choice of P and not only on m. Nearly tight lower and upper bounds follow in the case where m is even.

Original languageEnglish
Pages (from-to)957-974
Number of pages18
JournalDiscrete and Computational Geometry
Volume72
Issue number2
DOIs
StatePublished - Sep 2024

Keywords

  • 52C99
  • Concave chains
  • Convex chains
  • Convex polygons
  • Geometric graphs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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