Matroid Enumeration for Incidence Geometry

Yoshitake Matsumoto, Sonoko Moriyama, Hiroshi Imai, David Bremner

研究成果: Article査読

23 被引用数 (Scopus)

抄録

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.

本文言語English
ページ(範囲)17-43
ページ数27
ジャーナルDiscrete and Computational Geometry
47
1
DOI
出版ステータスPublished - 2012 1月
外部発表はい

ASJC Scopus subject areas

  • 理論的コンピュータサイエンス
  • 幾何学とトポロジー
  • 離散数学と組合せ数学
  • 計算理論と計算数学

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