Combinatorial face enumeration in arrangements and oriented matroids

Fukuda, Komei; Saito, Shigemasa; Tamura, Akihisa

doi:10.1016/0166-218x(91)90066-6

Cited by 20 publications

(12 citation statements)

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“…Furthermore, it is possible to decide for a given graph in polynomial time whether it is a tope graph or not [15], [16]. So, isomorphism classes of simple OMs can be generated if it is possible to generate tope graphs or at least a (not too large) superset of graphs.…”

Section: Tope Graphs and Single Element Extensionsmentioning

confidence: 99%

“…Note that our method is working with graph G and not with M. The main idea is to generate first all weak acycloidal signatures and then to test these signatures for being strong acycloidal signatures, finally for being localizations (again in polynomial time, see [15] and [16]). The tope graphs of the extended OMs are easily obtained from the localizations (see Section 3), and finally graph isomorphism checking leads to a set of representatives up to isomorphism.…”

Section: Generation Of Extensions By Reverse Searchmentioning

confidence: 99%

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Generation of Oriented Matroids—A Graph Theoretical Approach

Finschi¹,

Fukuda²

2002

Discrete Comput Geom

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We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for non-uniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore, we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs finally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.

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Section: Tope Graphs and Single Element Extensionsmentioning

confidence: 99%

Section: Generation Of Extensions By Reverse Searchmentioning

confidence: 99%

Generation of Oriented Matroids—A Graph Theoretical Approach

Finschi¹,

Fukuda²

2002

Discrete Comput Geom

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“…For results concerning inequalities and unimodality of the f -vector of zonotopes, see [17,18,19]. The minimal and maximal number of connected components of arrangements has been studied by Shnurnikov [30] in the cases of Euclidean, projective and Lobachevskiȋ spaces.…”

Section: Discussionmentioning

confidence: 99%

Manifold arrangements

Ehrenborg

Readdy

2014

Journal of Combinatorial Theory, Series A

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“…Other papers related to topes of oriented matroids can be seen in [11,16,17,30,45]. § 3 0 Acycloids, /^-Systems and Their Applications Let 3 be the set of topes of an oriented matroid M on E. An element of E is called a loop of M if it is not contained in the support of any tope, and the set of loops of M is denoted by E 0 .…”

Section: Y)mentioning

confidence: 99%

Topes of Oriented Matroids and Related Structures

Handa¹

1993

Publ. Res. Inst. Math. Sci.

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An oriented matroid can be viewed as a combinatorial abstraction of the facial incidence relations of the polyhedral cones induced by a finite arrangement of oriented hyperplanes in R d through the origin. "Topes" of an oriented matroid correspond to maximal polyhedral cones. This paper discusses three structures related to topes of oriented matroids, namely, acycloids, L by the subspaces {xeR w :xA e =0} (ee£), and also represents the facial incidence relations of the polyhedral cones. An oriented matroid is defined by a set of signed vectors satisfying certain axioms (face axioms} that are trivially satisfied by o (V). Besides this, an oriented matroid can be viewed as abstractions of many different concepts in linear space, see [9,10,26] for the basic theory and applications. "Topes" [27,39] of an oriented matroid correspond to maximal polyhedral cones in the above setting. Topes can be also considered an abstraction of some properties of acyclic reorientations of loopless directed graphs, and further-

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Combinatorial face enumeration in arrangements and oriented matroids

Cited by 20 publications

References 5 publications

Generation of Oriented Matroids—A Graph Theoretical Approach

Generation of Oriented Matroids—A Graph Theoretical Approach

Manifold arrangements

Topes of Oriented Matroids and Related Structures

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