Complete Enumeration of Small Realizable Oriented Matroids

Komei Fukuda, Hiroyuki Miyata, Sonoko Moriyama

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (Discrete Comput Geom 27:117-136, 2002) published the first database of oriented matroids including degenerate (i. e., non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points, and 5-dimensional configurations of 9 points. We also determine all possible combinatorial types of 5-polytopes with nine vertices.

Original languageEnglish
Pages (from-to)359-381
Number of pages23
JournalDiscrete and Computational Geometry
Issue number2
Publication statusPublished - 2013 Mar
Externally publishedYes

ASJC Scopus subject areas

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


Dive into the research topics of 'Complete Enumeration of Small Realizable Oriented Matroids'. Together they form a unique fingerprint.

Cite this