2007
DOI: 10.1016/j.jcta.2006.06.012
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Oriented matroids and complete-graph embeddings on surfaces

Abstract: We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable a… Show more

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Cited by 6 publications
(4 citation 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%
“…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%
“…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%
“…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].…”
mentioning
confidence: 99%