Lower bounds for electrical reduction on surfaces

Hsien Chih Chang, Marcos Cossarini, Jeff Erickson

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We strengthen the connections between electrical transformations and homotopy from the planar setting – observed and studied since Steinitz – to arbitrary surfaces with punctures. As a result, we improve our earlier lower bound on the number of electrical transformations required to reduce an n-vertex graph on surface in the worst case [SOCG 2016] in two different directions. Our previous Ω(n3/2) lower bound applies only to facial electrical transformations on plane graphs with no terminals. First we provide a stronger Ω(n2) lower bound when the planar graph has two or more terminals, which follows from a quadratic lower bound on the number of homotopy moves in the annulus. Our second result extends our earlier Ω(n3/2) lower bound to the wider class of planar electrical transformations, which preserve the planarity of the graph but may delete cycles that are not faces of the given embedding. This new lower bound follow from the observation that the defect of the medial graph of a planar graph is the same for all its planar embeddings.

Original languageEnglish (US)
Title of host publication35th International Symposium on Computational Geometry, SoCG 2019
EditorsGill Barequet, Yusu Wang
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771047
DOIs
StatePublished - Jun 1 2019
Event35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States
Duration: Jun 18 2019Jun 21 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume129
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Computational Geometry, SoCG 2019
CountryUnited States
CityPortland
Period6/18/196/21/19

Keywords

  • 2-flipping
  • Defect
  • Electrical transformation
  • Homotopy
  • Routing set
  • SPQR-tree
  • Smoothings
  • Tight
  • ∆Y-transformation

ASJC Scopus subject areas

  • Software

Fingerprint Dive into the research topics of 'Lower bounds for electrical reduction on surfaces'. Together they form a unique fingerprint.

  • Cite this

    Chang, H. C., Cossarini, M., & Erickson, J. (2019). Lower bounds for electrical reduction on surfaces. In G. Barequet, & Y. Wang (Eds.), 35th International Symposium on Computational Geometry, SoCG 2019 [25] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 129). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2019.25