Oriented matroids and complete-graph embeddings on surfaces

Bokowski, Jürgen; Pisanski, Tomaž

doi:10.1016/j.jcta.2006.06.012

Cited by 6 publications

(4 citation statements)

References 31 publications

(33 reference statements)

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“…The problem of counting some special triplets of vertices in complete graph drawings is addressed in [16,2,7]. The problem of drawing complete graphs on general surfaces is addressed from a different viewpoint (though also related to oriented matroids) in [11]. Equivalence classes for general graph drawings with the same sets of pairs of edges that cross are addressed in [17,24,20].…”

Section: Introductionmentioning

confidence: 99%

Complete Graph Drawings Up to Triangle Mutations

Gioan

2005

Graph-Theoretic Concepts in Computer Science

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The main result of the paper can be stated in the following way: a complete graph drawing in the sphere, where two edges have at most one common point, which is either a crossing or a common endpoint, and no three edges share a crossing, is determined by the circular ordering of edges at each vertex, or equivalently by the set of pairs of edges that cross, up to homeomorphism and a sequence of triangle mutations. A triangle mutation (or switch, or flip) consists in passing an edge across the intersection of two other edges, assuming the three edges cross each other and the region delimited by the three edges has an empty intersection with the drawing. Equivalently, the result holds for a drawing in the plane assuming the drawing is given with a pair of edges indicating where the unbounded region is. The proof is constructive, based on an inductive algorithm that adds vertices and their incident edges one by one (actually yielding an extra property for the sequence of triangle mutations). This result generalizes Ringel's theorem on uniform pseudoline arrangements (or rank 3 uniform oriented matroids) to complete graph drawings. We also apply this result to plane projections (or visualizations) of a geometric spatial complete graph, in terms of the rank 4 uniform oriented matroid defined by its vertices. Independently, we prove that, for a complete graph drawing, the set of pairs of edges that cross determine, by first order logic formulas, the circular ordering of edges at each vertex, as well as further information.

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Section: Introductionmentioning

confidence: 99%

Complete Graph Drawings Up to Triangle Mutations

Gioan

2005

Graph-Theoretic Concepts in Computer Science

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“…For example, the three (geometric) (9 3 ) configurations (9 3 ) 1 , (9 3 ) 2 , and (9 3 ) 3 , compare [8], Figure 2.2.1, have different cycle length vectors (6,6,14,14,14), (3,3,3,3,3,3,3,33), and (3, 3, 3, 14, 14, 14), respectively. When we have another geometric version of one of these drawings, we can tell via the cycle length vector which version we have.…”

Section: Mutation Class Of a Topological Configurationmentioning

confidence: 99%

“…Before we write the next theorem with properties of configuration manifolds, we wish to mention [6], i.e., another paper with a link between pseudo-line arrangements and topological graph theory. 3.…”

Section: Properties Of the Configuration Manifoldmentioning

confidence: 99%

A manifold associated to a topological (n_k) configuration

Bokowski

Strausz

2014

Ars Math. Contemp.

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In the study of topological (n k) configurations in the projective plane we have n pseudolines and n points. Precisely k of these points are incident with each pseudo-line and, vice versa, precisely k pseudo-lines are incident with each point. We describe a non-orientable 2-manifold associated in a natural way to a topological (n k) configuration in the projective plane. Apart from being interesting in its own right, this manifold defines an equivalence class within topological configurations. It can be used to distinguish topological (n k) configurations.

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“…9 Quasi-topological incidence structures as systems of curves on surfaces A curve arrangement is a collection of simple closed curves on a given surface such that each pair of curves (c i , c j ), i = j , has at most one point in common at which they cross transversely. In addition the arrangement should be cellular, i.e., the complement of the curves is a union of open discs; see J. Bokowski and T. Pisanski [6].…”

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confidence: 99%

Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces

et al. 2017

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It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topological. In this paper we provide a new topological representation by using and essentially generalizing the topological representation of oriented matroids in rank 3. These representations can also be interpreted as curve arrangements on surfaces.In particular, we generalize the notion of a pseudoline arrangement to the notion of a quasiline arrangement by relaxing the condition that two pseudolines meet exactly once and show that every combinatorial configuration can be realized as a quasiline arrangement in the real projective plane. We also generalize well-known tools from pseudoline arrangements such as sweeps or wiring diagrams. A quasiline arrangement with selected vertices belonging to the configuration can be viewed as a map on a closed surface. Such a map can be used to distinguish between two "distinct" realizations of a combinatorial configuration as a quasiline arrangement.

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Oriented matroids and complete-graph embeddings on surfaces

Cited by 6 publications

References 31 publications

Complete Graph Drawings Up to Triangle Mutations

Complete Graph Drawings Up to Triangle Mutations

A manifold associated to a topological (n_k) configuration

Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces

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