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The flow lattice of oriented matroids
 CONTRIBUTIONS TO DISCRETE MATHEMATICS
"... Recently Hochstättler and Nesetril introduced the flow lattice of an oriented matroid as generalization of the lattice of all integer flows of a digraph or more general a regular matroid. This lattice is defined as the integer hull of the characteristic vectors of signed circuits. Here, we character ..."
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Recently Hochstättler and Nesetril introduced the flow lattice of an oriented matroid as generalization of the lattice of all integer flows of a digraph or more general a regular matroid. This lattice is defined as the integer hull of the characteristic vectors of signed circuits. Here, we characterize the flow lattice of oriented matroids that are uniform or have rank 3 with a particular focus on the dimension of the lattice and construct a basis consisting of directed circuits. For general oriented matroids we introduce a 2sum and decompose oriented matroids into 3connected parts. We show how to determine the dimension of the lattice of 2sums and conclude with some questions based on extensive experiments on small oriented matroids with connectivity at least 3.
Balanced Signings of Oriented Matroids and Chromatic Number
 FLOW LATTICE OF ORIENTED MATROIDS 19
, 2003
"... This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the sign p ..."
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Cited by 4 (3 self)
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This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the sign patterns of (signed) circuits (or cocircuits). The result is a natural invariant of an oriented matroid. The invariant can be viewed as a `discrepancy in ratio' of a pseudohyperplane arrangement, and thus should be of interest to geometers. The main theorem answers a question raised in [7]. We first state the result and some consequences, using a minimal set of definitions. Detailed definitions appear in Section 2. It is convenient to view an oriented matroid to be a matroid in which every circuit C (and cocircuit B) has been partitioned C = C C  , (and B = B subject to a standard orthogonality condition. We regard each bipartition as an unordered pair }, where one of the parts may be empty. For E(O), the reorientation I of is the new oriented matroid obtained from by repartitioning each circuit C (and cocircuit B) according to the rules } ## {C C), C  C)} and } ## {B B), B  B)}
VapnikChervonenkis dimension and (pseudo)hyperplane arrangements
, 1997
"... An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let ..."
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An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let R be the collection of all these subsets. We investigate and characterize the range spaces corresponding to simple arrangements of pseudohyperplanes in this way; such range spaces are called pseudogeometric, and they have the property that the cardinality of R is maximum for the given VCdimension. In general, such range spaces are called maximum, and we show that the number of ranges R ∈ R for which also X −R ∈ R, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and ‘small ’ subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniform oriented matroids: a range space (X, R) naturally corresponds to a uniform oriented matroid of rank X  − d if and only if its VCdimension is d, R ∈ R implies X − R ∈ R and R  is maximum under these conditions.
Biased Graphs: IV. Geometrical Realizations
, 2000
"... . Biased graphs abstract gain graphs, which are graphs whose oriented edges are labelled invertibly from a group. A biased, hence gain, graph has two natural matroids: the bias matroid G and the (complete) lift matroid L (or L 0 ). If the gain group is contained in the multiplicative group of a s ..."
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. Biased graphs abstract gain graphs, which are graphs whose oriented edges are labelled invertibly from a group. A biased, hence gain, graph has two natural matroids: the bias matroid G and the (complete) lift matroid L (or L 0 ). If the gain group is contained in the multiplicative group of a skew eld, the bias matroid is representable by vectors over and also in several ways by hyperplanes; two of these representations generalize the classical Ceva and Menelaos theorems, while one dualizes the vector representation. The vector representation is unique for `full' but not for all gain graphs. The dualizing hyperplane representation can be abstracted away from elds to a form of equational logic. If the gain group is contained in the additive group of a skew eld, then the complete lift matroid is representable by vectors and dually by linear, projective, and ane hyperplanes. The representations are unique for the complete lift but not for the incomplete lift matroid. Intro...
On Radon's theorem and representations of separoids
 Charles University at Prague
, 2003
"... Separoids a natural generalization of oriented matroids are symetric relations 2 F defined on pairs of disjoint subsets which are closed, as filters, by the natural partial order D); they encode the separation structure of the families convex sets as follows: a pair of disjoi ..."
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Separoids a natural generalization of oriented matroids are symetric relations 2 F defined on pairs of disjoint subsets which are closed, as filters, by the natural partial order D); they encode the separation structure of the families convex sets as follows: a pair of disjoint subsets A, B is a Radon partition, denoted as A+B, i# their convex hulls intersect; otherwise, we say that they are separated hence the name of the structure. We will show here that every separoid S can be represented (realized) with a family of convex sets ; that is, given an abstract (combinatorial) separoid S we will construct a family of convex sets whose separation structure (whose Radon partitions) are exactly those of the separoid S.
On the topology and geometric construction of oriented matroids and convex polytopes
 Transactions Amer. Math. Soc
, 1991
"... Abstract. This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M. We focus our attention on point configurations in general position, and as the main result we prove that the reali ..."
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Abstract. This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M. We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank 3 oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces 32(M) are pathconnected. We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes. 1.
Boundary complexes of convex polytopes cannot be characterized locally
 BULL. LONDON MATH. SOC
, 1987
"... It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the setup of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial spher ..."
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It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the setup of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial sphere is polytopal. The proof is based on a construction technique for rigid chirotopes. These correspond, in the realizable case, to convex polytopes whose internal combinatorial structure is completely determined by its face lattice. So, a rigid chirotope is realizable over a field F if and only if its facelattice is Fpolytopal. Furthermore we prove that for every proper subfield F of the field A of real algebraic numbers there exists a 6polytope which is not realizable over F.
On Delaunay Oriented Matroids For Convex Distance Functions.
 University of Saskatoon
, 1995
"... For any finite point set S in E d , an oriented matroid DOM(S) can be defined in terms of how S is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation of S and is realizable, because of the lifting property of Delaunay triangulations. We prove that ..."
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For any finite point set S in E d , an oriented matroid DOM(S) can be defined in terms of how S is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation of S and is realizable, because of the lifting property of Delaunay triangulations. We prove that the same construction of a Delaunay oriented matroid can be performed with respect to any smooth, strictly convex distance function in the plane E 2 (Theorem 3.5). For these distances, the existence of a Delaunay oriented matroid cannot follow from a lifting property, because Delaunay triangulations might be nonregular (Theorem 4.2(i). This is related to the fact that the Delaunay oriented matroid can be nonrealizable (Theorem 4.2(ii) ). Keywords : oriented matroid, Delaunay triangulation, Voronoi diagram 1 Introduction In this paper we describe a link between the Delaunay triangulation of a finite point set S in the Euclidean dspace E d and a certain oriented matroid DOM(S) of ra...
A LatticeTheoretical Characterization of Oriented Matroids
 EUROP. J. COMBINATORICS
, 1997
"... If P is the big face lattice of the covectors of an oriented matroid, it is well known that the zero map is a coverpreserving, orderreversing surjection onto the geometric lattice of the underlying (unoriented) matroid. In this paper we give a (necessary and) sufficient condition for such maps ..."
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If P is the big face lattice of the covectors of an oriented matroid, it is well known that the zero map is a coverpreserving, orderreversing surjection onto the geometric lattice of the underlying (unoriented) matroid. In this paper we give a (necessary and) sufficient condition for such maps to come from the face lattice of an oriented matroid.
Linear Programming Duality and Morphisms
, 1998
"... In this paper we investigate the class NP " coNP (or the class of problems permitting a good characterisation) from the point of view of morphisms of oriented matroids. We prove several morphismduality theorems for oriented matroids. These generalize LPduality (in form of Farkas' Lemma) and Minty ..."
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In this paper we investigate the class NP " coNP (or the class of problems permitting a good characterisation) from the point of view of morphisms of oriented matroids. We prove several morphismduality theorems for oriented matroids. These generalize LPduality (in form of Farkas' Lemma) and Minty's Painting Lemma. Moreover, we characterize all morphism duality theorems, thus proving the essential unicity of Farkas' Lemma. This research helped to isolate perhaps the most natural definition of strong maps for oriented matroids.