Hanani-Tutte and Related Results

Schaefer, Marcus

doi:10.1007/978-3-642-41498-5_10

Cited by 18 publications

(27 citation statements)

References 54 publications

(79 reference statements)

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“…We refer the reader to [17,22,23,24] for more background on the family of Hanani-Tutte theorems, but suffice it to say that strong variants are still rather rare, so we consider the current result as important evidence that Hanani-Tutte is a viable route to graph-drawing questions.…”

Section: Introductionmentioning

confidence: 97%

“…On the other hand, strong Hanani-Tutte theorems are often algorithmic. Theorem 1 yields a very simple algorithm for radial planarity testing (described in Section 5) which is based on solving a system of linear equations over Z/2Z, see also [23,Section 1.4]. Our algorithm runs in time O(n 2ω ), where n = |V (G)| and O(n ω ) is the complexity of multiplication of two square n × n matrices.…”

Section: Introductionmentioning

confidence: 99%

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Hanani-Tutte for Radial Planarity II

Fulek¹,

Pelsmajer²,

Schaefer³

2016

Lecture Notes in Computer Science

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Abstract. A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, . . . , C k with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. A pair of edges e and f in a graph is independent if e and f do not share a vertex. We show that a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Hanani-Tutte for Radial Planarity II

Fulek¹,

Pelsmajer²,

Schaefer³

2016

Lecture Notes in Computer Science

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“…This point of view leads to challenging open questions (see, for example, [23]); even in two dimensions-for surfaces-the problem is not understood well. (See [31] for a survey of what we do know. )…”

Section: Introductionmentioning

confidence: 99%

“…(For background and variants of the weak Hanani-Tutte theorem, see [31].) For xmonotone drawings this translates to the claim that if there is an x-monotone drawing in which every pair of edges crosses an even number of times, then there is an x-monotone embedding with the same vertex locations.…”

Section: Introductionmentioning

confidence: 99%

Hanani–Tutte, Monotone Drawings, and Level-Planarity

Fulek

Pelsmajer

Schaefer

et al. 2012

Thirty Essays on Geometric Graph Theory

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A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph has an x-monotone drawing in which every pair of edges crosses an even number of times, then the graph has an x-monotone embedding in which the x-coordinates of all vertices are unchanged. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross each other oddly. This answers a question posed by Pach and Tóth. We show that a further strengthening to a "removing even crossings" lemma is impossible by separating monotone versions of the crossing and the odd crossing number.Our results extend to level-planarity, which is a well-studied generalization of x-monotonicity. We obtain a new and simple algorithm to test level-planarity in quadratic time, and we show that x-monotonicity of edges in the definition of level-planarity can be relaxed.

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“…This algorithm (sketched in Section 4) is based on solving a system of linear equations over Z 2 , see also [19,Section 1.4]. Thus, our algorithm runs in time O(|V (G)| 2ω ), where O(n ω ) is the complexity of multiplication of two square n × n matrices.…”

Section: Introductionmentioning

confidence: 99%

Hanani-Tutte for Radial Planarity

Fulek¹,

Pelsmajer²,

Schaefer³

2017

JGAA

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Abstract. A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, . . . , C k with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.

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Hanani-Tutte and Related Results

Cited by 18 publications

References 54 publications

Hanani-Tutte for Radial Planarity II

Hanani-Tutte for Radial Planarity II

Hanani–Tutte, Monotone Drawings, and Level-Planarity

Hanani-Tutte for Radial Planarity

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