Removing even crossings

Pelsmajer, Michael J.; Schaefer, Marcus; Štefankovič, Daniel

doi:10.1016/j.jctb.2006.08.001

Cited by 35 publications

(73 citation statements)

References 12 publications

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“…In a recent paper [34] we introduced the name "strong Hanani-Tutte theorem" to distinguish it from a weaker version that is also often called the Hanani-Tutte theorem in the literature.…”

Section: The Hanani-tutte Theorem In the Planementioning

confidence: 99%

Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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We investigate under what conditions crossings of adjacent edges and pairs of edges crossing an even number of times are unnecessary when drawing graphs. This leads us to explore the Hanani-Tutte theorem and its close relatives, emphasizing the intuitive geometric content of these results. The Hanani-Tutte Theorem in The PlaneIn 1934, Hanani [15] published a paper which-in passing-established the following result:Any drawing of a K 5 or a K 3,3 contains two independent edges crossing each other oddly. 1Since by Kuratowski's theorem every non-planar graph contains a subdivision of K 5 or K 3,3 , Hanani's observation implies that any drawing of a nonplanar graph contains two vertex-disjoint paths that cross an odd number of times and therefore two independent edges that cross oddly-one in each path. This consequence was first explicitly stated by Tutte [49]. 2 1 The result can be hard to find even if one reads German. It is stated as (1) on page 137 of the article and mainly an application of methods developed by Flores [20, 62. Kolloqium].2 The theorem is generally known as the Hanani-Tutte theorem, though Levow [17] calls it the "van Kampen-Shapiro-Wu characterization of planar graphs" emphasizing the parallel history of the theorem in algebraic topology (ignoring Flores, however). In a recent paper [34] we introduced the name "strong Hanani-Tutte theorem" to distinguish it from a weaker version that is also often called the Hanani-Tutte theorem in the literature.

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“…In a recent paper [34] we introduced the name "strong Hanani-Tutte theorem" to distinguish it from a weaker version that is also often called the Hanani-Tutte theorem in the literature.…”

Section: The Hanani-tutte Theorem In the Planementioning

confidence: 99%

Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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“…1 We give a new proof of this result as Theorem 1 in Section 2, which continues an elementary topological approach similar to earlier papers on the Hanani-Tutte theorem, e.g. [15].…”

Section: Introductionmentioning

confidence: 76%

“…We approach Theorem 1 in the spirit of earlier papers on the Hanani-Tutte theorem, e.g. [15]. The proof, which is omitted, repeatedly makes use of a simple topological observation: suppose we are given two curves (not necessarily monotone) starting at x = x 1 and ending at x = x 2 which lie entirely between x 1 and x 2 .…”

Section: Theorem 1 (Pach Tóthmentioning

confidence: 99%

“…For example, we know that ocr(G) = cr(G) for ocr(G) ≤ 3 [16] and iocr(G) = cr(G) for iocr(G) ≤ 2 [17]. Pach and Tóth showed that cr(G) ≤ ocr(G) 2 [11,15]. The core step in this result is a "removing even crossings" lemma, in this particular case: if G is drawn in the plane and E 0 is the set of its even edges, then G can be redrawn so that all edges in E 0 are free of crossings.…”

Section: Monotone Crossing Numbersmentioning

confidence: 99%

“…The truth of this result was left as an open problem by Pach and Tóth.We prove it as Theorem 2 in Section 3. The extension of this result in the spirit of [11,15] is not possible, which is proved in Section 4.…”

Section: Introductionmentioning

confidence: 96%

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Hanani-Tutte and Monotone Drawings

Fulek

Pelsmajer

Schaefer

et al. 2011

Graph-Theoretic Concepts in Computer Science

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Abstract.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 oddly. This answers a question posed by Pach and Tóth. Moreover, we show that an extension of this result for graphs with non-adjacent pairs of edges crossing oddly fails even if there exists only one such pair in a graph.

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Finding Minors in Graphs with a Given Path Structure

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Pelsmajer

Ramamurthi

2014

Journal of Graph Theory

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Given graphs G and H with V(H)⊆V(G), suppose that we have a u,v‐path Puv in G for each edge uv in H. There are obvious additional conditions that ensure that G contains H as a rooted subgraph, subdivision, or immersion; we seek conditions that ensure that G contains H as a rooted minor or minor. This naturally leads to studying sets of paths that form an H‐immersion, with the additional property that paths that contain the same vertex must have a common endpoint. We say that H is contractible if, whenever G contains such an H‐immersion, G must also contain a rooted H‐minor. We show, for example, that forests, cycles, K4, and K1, 1, 3 are contractible, but that graphs that are not 6‐colorable and graphs that contain certain subdivisions of K2, 3 are not contractible.

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Removing even crossings

Cited by 35 publications

References 12 publications

Hanani-Tutte and Related Results

Hanani-Tutte and Related Results

Hanani-Tutte and Monotone Drawings

Finding Minors in Graphs with a Given Path Structure

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