Convex drawings of the complete graph: topology meets geometry

Arroyo, Alan; McQuillan, Dan; Richter, R. Bruce; Salazar, Gelasio

doi:10.26493/1855-3974.2134.ac9

Cited by 4 publications

(4 citation statements)

References 18 publications

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“…Using analogous techniques as in the proof of Theorem 1, we show the following. Actually we prove this last bound not only for spherical geodesic drawings, but for the more general class of convex drawings [7,8]. A drawing D of K n in the sphere is convex if, for every 3-cycle C, there is a closed disc ∆ bounded by C with the following property: for any two vertices u, v contained in ∆, the edge uv is contained in ∆.…”

Section: Our Main Resultsmentioning

confidence: 82%

Closing in on Hill's Conjecture

Balogh¹,

Lidický²,

Salazar³

2019

SIAM J. Discrete Math.

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Section: Our Main Resultsmentioning

confidence: 82%

Closing in on Hill's Conjecture

Balogh¹,

Lidický²,

Salazar³

2019

SIAM J. Discrete Math.

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“…Clearly f-convex drawings are h-convex and h-convex drawings are convex. Evidence is given in [9] to support the conjecture that every crossingminimal drawing of K n is convex. A polynomial-time algorithm recognizing h-convexity follows from their result that a drawing of K n is h-convex if and only if it does not contain as a subdrawing any of the three drawings (two of K 5 and one of K 6 ) shown in Figure 7.…”

Section: H-convex and Pseudospherical Drawingsmentioning

confidence: 99%

“…A polynomial-time algorithm recognizing h-convexity follows from their result that a drawing of K n is h-convex if and only if it does not contain as a subdrawing any of the three drawings (two of K 5 and one of K 6 ) shown in Figure 7. We do not need this result here and there is little overlap of this work with [9]. 3) is a routine exercise.…”

Section: H-convex and Pseudospherical Drawingsmentioning

confidence: 99%

Extending drawings of complete graphs into arrangements of pseudocircles

Arroyo¹,

Richter

Sunohara

2020

Preprint

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Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of K n , we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph G is a drawing in the unit sphere S 2 in which the vertices of G are represented as points-no three on a great circle-and the edges of G are shortest-arcs in S 2 connecting pairs of vertices. Such a drawing has three properties: every edge e is contained in a simple closed curve γ e such that the only vertices in γ e are the ends of e; if e = f , then γ e ∩ γ f has precisely two crossings; and if e = f , then e intersects γ f at most once, either a crossing or an end of e. We use these three properties to define a pseudospherical drawing of G. Our main result is that, for the complete graph, these three properties are equivalent to the same three properties but with "exactly two crossings" replaced by "at most two crossings".

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“…The problem of counting edge crossings in monotone drawings of the complete graph is addressed [8]. Convex drawings of the complete graph are addressed in [6]. More specific references are also given along the paper.…”

Section: Introductionmentioning

confidence: 99%

Complete Graph Drawings up to Triangle Mutations

Gioan¹

2022

Discrete Comput Geom

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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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Convex drawings of the complete graph: topology meets geometry

Cited by 4 publications

References 18 publications

Closing in on Hill's Conjecture

Closing in on Hill's Conjecture

Extending drawings of complete graphs into arrangements of pseudocircles

Complete Graph Drawings up to Triangle Mutations

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