A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion

Sarkaria, K. S.

doi:10.1007/bf02773426

Cited by 34 publications

(20 citation statements)

References 11 publications

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“…This is why we believe that Sarkaria's proof cannot be fixed along the lines described in his paper. 7 Sarkaria [41] also claims this result, however we were not able to verify it.…”

Section: Algorithmic and Algebraic Aspectsmentioning

confidence: 69%

“…These problems were first addressed in the sixties, culminating in the linear-time algorithm by Hopcroft and Tarjan. The HananiTutte theorem offers an alternative algorithmic approach to planarity testing along two separate routes: one practical, the tree approach, based on work of 6 Sarkaria [41] in 1991 claimed the same result. His proof contains several flaws: The redrawing suggested in his Figure 4 (page 82) introduces odd crossings between β and edges that end between α and β.…”

Section: Algorithmic and Algebraic Aspectsmentioning

confidence: 97%

“…However, solving quadratic systems of equations is NP-complete, so Wu's approach does not lead to an efficient solution. Sarkaria [41] claims that there is a "onedimensional version of Whitney's trick by means of which any graph [G which has an i-even drawing] can be, step by step embedded in R 2 ." Unfortunately, his version of the Whitney trick for n = 1 is fatally flawed as explained in an earlier footnote.…”

Section: Surfacesmentioning

confidence: 99%

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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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“…This is why we believe that Sarkaria's proof cannot be fixed along the lines described in his paper. 7 Sarkaria [41] also claims this result, however we were not able to verify it.…”

Section: Algorithmic and Algebraic Aspectsmentioning

confidence: 69%

Section: Algorithmic and Algebraic Aspectsmentioning

confidence: 97%

Section: Surfacesmentioning

confidence: 99%

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Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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“…We also omit the details. K 3,3 ) are homeomorphic to the closed connected orientable surfaces of genus 6 and 4, respectively [13]. Then, for an automorphism…”

Section: Proof Of Proposition 18 (1)mentioning

confidence: 99%

Symmetries of spatial graphs and Simon invariants

Nikkuni

Taniyama

2009

Fund. Math.

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Abstract. An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3 + 3 vertices in detail, and determine necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.1. Introduction. Throughout this paper we work in the piecewise linear category. Let L = J 1 ∪ J 2 be an ordered and oriented 2-component link in the unit 3-sphere S 3 . Unless otherwise stated, the links in this paper will be ordered and oriented. A link L is said to be component preserving achiral (CPA) if there exists an orientation-reversing self-homeomorphism ϕ of S 3 such that ϕ(J 1 ) = J 1 and ϕ(J 2 ) = J 2 , and component switching achiral (CSA) if there exists an orientation-reversing self-homeomorphism ϕ of S 3 such that ϕ(J 1 ) = J 2 and ϕ(J 2 ) = J 1 [3]. If L is either CPA or CSA, then L is said to be achiral. Note that L may be both CPA and CSA (a trivial link, for example). The following was shown by Kirk-Livingston.

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“…In this paper we want to relax the embedding condition using Z 2 -homology, that is, we are only interested in the parity of the number of crossings between independent edges; in terms of algebraic topology we are studying the "mod 2 homology of the deleted product of the graph" [1]. We say a graph Z 2 -embeds in S if it can be drawn in S so that every pair of independent edges crosses evenly.…”

Section: Introductionmentioning

confidence: 99%