Matroid Enumeration for Incidence Geometry

Yoshitake Matsumoto, Sonoko Moriyama, Hiroshi Imai, David Bremner

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)

Abstract

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points-lines-planes conjecture and the so-called Sylvester-Gallai type problems derived from the Sylvester-Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh-Seymour on ≤11 points in ℝ3 and that of Motzkin on ≤12 lines in ℝ2, extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.

Original languageEnglish
Pages (from-to)17-43
Number of pages27
JournalDiscrete and Computational Geometry
Volume47
Issue number1
DOIs
Publication statusPublished - 2012 Jan

Keywords

  • Incidence geometry
  • Matroids
  • Oriented matroids

ASJC Scopus subject areas

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

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