Degenerate crossing number and signed reversal distance
The degenerate crossing number of a graph is the minimum number of transverse crossings among all its drawings, where edges are represented as simple arcs and multiple edges passing through the same point are counted as a single crossing. Interpreting each crossing as a cross-cap induces an embeddin...
Saved in:
Published in | arXiv.org |
---|---|
Main Authors | , , |
Format | Paper |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
22.08.2023
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Abstract | The degenerate crossing number of a graph is the minimum number of transverse crossings among all its drawings, where edges are represented as simple arcs and multiple edges passing through the same point are counted as a single crossing. Interpreting each crossing as a cross-cap induces an embedding into a non-orientable surface. In 2007, Mohar showed that the degenerate crossing number of a graph is at most its non-orientable genus and he conjectured that these quantities are equal for every graph. He also made the stronger conjecture that this also holds for any loopless pseudotriangulation with a fixed embedding scheme. In this paper, we prove a structure theorem that almost completely classifies the loopless 2-vertex embedding schemes for which the degenerate crossing number equals the non-orientable genus. In particular, we provide a counterexample to Mohar's stronger conjecture, but show that in the vast majority of the 2-vertex cases, the conjecture does hold. The reversal distance between two signed permutations is the minimum number of reversals that transform one permutation to the other one. If we represent the trajectory of each element of a signed permutation under successive reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme with degenerate crossings. Our main result is proved by leveraging this connection and a classical result in genome rearrangement (the Hannenhali-Pevzner algorithm) and can also be understood as an extension of this algorithm when the reversals do not necessarily happen in a monotone order. |
---|---|
AbstractList | The degenerate crossing number of a graph is the minimum number of transverse crossings among all its drawings, where edges are represented as simple arcs and multiple edges passing through the same point are counted as a single crossing. Interpreting each crossing as a cross-cap induces an embedding into a non-orientable surface. In 2007, Mohar showed that the degenerate crossing number of a graph is at most its non-orientable genus and he conjectured that these quantities are equal for every graph. He also made the stronger conjecture that this also holds for any loopless pseudotriangulation with a fixed embedding scheme. In this paper, we prove a structure theorem that almost completely classifies the loopless 2-vertex embedding schemes for which the degenerate crossing number equals the non-orientable genus. In particular, we provide a counterexample to Mohar's stronger conjecture, but show that in the vast majority of the 2-vertex cases, the conjecture does hold. The reversal distance between two signed permutations is the minimum number of reversals that transform one permutation to the other one. If we represent the trajectory of each element of a signed permutation under successive reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme with degenerate crossings. Our main result is proved by leveraging this connection and a classical result in genome rearrangement (the Hannenhali-Pevzner algorithm) and can also be understood as an extension of this algorithm when the reversals do not necessarily happen in a monotone order. |
Author | Fuladi, Niloufar de Mesmay, Arnaud Hubard, Alfredo |
Author_xml | – sequence: 1 givenname: Niloufar surname: Fuladi fullname: Fuladi, Niloufar – sequence: 2 givenname: Alfredo surname: Hubard fullname: Hubard, Alfredo – sequence: 3 givenname: Arnaud surname: de Mesmay fullname: de Mesmay, Arnaud |
BookMark | eNqNyk0KwjAQQOEgClbtHQKuCzFpTPf-4AHcl9iMIaVOdKb1_LrwAK7e4nsrMceMMBOFNmZXNbXWS1Ey90opvXfaWlMId4QICORHkB1l5oRR4vS4AUmPQXKKCEESvIHYDzIkHj12sBGLux8Yyl_XYns-XQ-X6kn5NQGPbZ8nwi-1urHW1a5ujPnv-gBwyjfl |
ContentType | Paper |
Copyright | 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
Copyright_xml | – notice: 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
DBID | 8FE 8FG ABJCF ABUWG AFKRA AZQEC BENPR BGLVJ CCPQU DWQXO HCIFZ L6V M7S PIMPY PQEST PQQKQ PQUKI PTHSS |
DatabaseName | ProQuest SciTech Collection ProQuest Technology Collection Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central ProQuest Central Essentials ProQuest Central Technology Collection ProQuest One Community College ProQuest Central SciTech Premium Collection (Proquest) (PQ_SDU_P3) ProQuest Engineering Collection Engineering Database Publicly Available Content Database ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Academic ProQuest One Academic UKI Edition Engineering Collection |
DatabaseTitle | Publicly Available Content Database Engineering Database Technology Collection ProQuest Central Essentials ProQuest One Academic Eastern Edition ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Technology Collection ProQuest SciTech Collection ProQuest Central ProQuest Engineering Collection ProQuest One Academic UKI Edition ProQuest Central Korea Materials Science & Engineering Collection ProQuest One Academic Engineering Collection |
DatabaseTitleList | Publicly Available Content Database |
Database_xml | – sequence: 1 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
DeliveryMethod | fulltext_linktorsrc |
Discipline | Physics |
EISSN | 2331-8422 |
Genre | Working Paper/Pre-Print |
GroupedDBID | 8FE 8FG ABJCF ABUWG AFKRA ALMA_UNASSIGNED_HOLDINGS AZQEC BENPR BGLVJ CCPQU DWQXO FRJ HCIFZ L6V M7S M~E PIMPY PQEST PQQKQ PQUKI PTHSS |
ID | FETCH-proquest_journals_28557474833 |
IEDL.DBID | 8FG |
IngestDate | Thu Oct 10 17:22:59 EDT 2024 |
IsOpenAccess | true |
IsPeerReviewed | false |
IsScholarly | false |
Language | English |
LinkModel | DirectLink |
MergedId | FETCHMERGED-proquest_journals_28557474833 |
OpenAccessLink | https://www.proquest.com/docview/2855747483?pq-origsite=%requestingapplication% |
PQID | 2855747483 |
PQPubID | 2050157 |
ParticipantIDs | proquest_journals_2855747483 |
PublicationCentury | 2000 |
PublicationDate | 20230822 |
PublicationDateYYYYMMDD | 2023-08-22 |
PublicationDate_xml | – month: 08 year: 2023 text: 20230822 day: 22 |
PublicationDecade | 2020 |
PublicationPlace | Ithaca |
PublicationPlace_xml | – name: Ithaca |
PublicationTitle | arXiv.org |
PublicationYear | 2023 |
Publisher | Cornell University Library, arXiv.org |
Publisher_xml | – name: Cornell University Library, arXiv.org |
SSID | ssj0002672553 |
Score | 3.4942687 |
SecondaryResourceType | preprint |
Snippet | The degenerate crossing number of a graph is the minimum number of transverse crossings among all its drawings, where edges are represented as simple arcs and... |
SourceID | proquest |
SourceType | Aggregation Database |
SubjectTerms | Algorithms Embedding Permutations |
Title | Degenerate crossing number and signed reversal distance |
URI | https://www.proquest.com/docview/2855747483 |
hasFullText | 1 |
inHoldings | 1 |
isFullTextHit | |
isPrint | |
link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1NS8QwEB10i-DNT_xYl4Beg22TtulJUFsXYZdFFPa2pM14El3b7tXf7kzs6kHYYwgkIQnzZiYv8wCu8NWgIWCQCpNM6sgqWeUulAZrZylesZGXb5tM0_GLfpwn8z7h1va0yrVN9IbafdScI7-OTZKQ66uNull-SlaN4tfVXkJjG4KIK-HxT_Hy4TfHEqcZeczqn5n12FHuQTCzS2z2YQvfD2DHUy7r9hCyewrYueZzh8JjFYGI-BHoEBTeC2ZWoBNcYqlp7Ztw7OnRER3BZVk8343lerZFfx_axd_q1TEMKLDHE2YVKdQmxJp8BF0lkc21qjIdYhyaOnX5KQw3jXS2ufscdlkanfOfcTyEQdes8IIAtKtGfpdGENwW09kTtSZfxTcVbXrP |
link.rule.ids | 786,790,12792,21416,33406,33777,43633,43838 |
linkProvider | ProQuest |
linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1NS8QwEB20RfTmJ36sGtBrsG2SNj0J6i5Vd8siK-ytpM14knVt6_93Urt6EPYcSEIS5s2bPOYBXOObRk3AwAWqhMvQCF6mNuAaK2uIr5iws2-b5HH2Kp_mat4X3JpeVrmKiV2gth-Vq5HfRFopSn2lFrfLT-5co9zvam-hsQm-a7mpPfDvhvn05bfKEsUJ5cziX6Dt0GO0C_7ULLHegw1c7MNWJ7qsmgNIHoiyu67PLbIOrQhG2I9FByOCz5y2Ai1zTZbqxrwz63I9uqRDuBoNZ_cZX61W9C-iKf72L47AI2qPx05XJFDqACvKEmSpQpNKUSYywCjQVWzTExism-l0_fAlbGezybgYP-bPZ7DjjNJdNTSKBuC19ReeE5y25UV_Zt9WMHxb |
openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Degenerate+crossing+number+and+signed+reversal+distance&rft.jtitle=arXiv.org&rft.au=Fuladi%2C+Niloufar&rft.au=Hubard%2C+Alfredo&rft.au=de+Mesmay%2C+Arnaud&rft.date=2023-08-22&rft.pub=Cornell+University+Library%2C+arXiv.org&rft.eissn=2331-8422 |