License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.23
URN: urn:nbn:de:0030-drops-59152
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5915/
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Borradaile, Glencora ; Chambers, Erin Wolf ; Fox, Kyle ; Nayyeri, Amir

Minimum Cycle and Homology Bases of Surface Embedded Graphs

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LIPIcs-SoCG-2016-23.pdf (0.6 MB)

Abstract

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the 1-dimensional (Z_2)-homology classes) of an undirected graph embedded on an orientable surface of genus g. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis.

For the minimum cycle basis problem, we give a deterministic O(n^omega + 2^2g n^2)-time algorithm. The best known existing algorithms for surface embedded graphs are those for general sparse graphs: an O(n^omega) time Monte Carlo algorithm [Amaldi et. al., ESA'09] and a deterministic O(n^3) time algorithm [Mehlhorn and Michail, TALG'09]. For the minimum homology basis problem, we give an O(g^3 n log n)-time algorithm, improving on existing algorithms for many values of g and n.

BibTeX - Entry

@InProceedings{borradaile_et_al:LIPIcs:2016:5915,
  author =	{Glencora Borradaile and Erin Wolf Chambers and Kyle Fox and Amir Nayyeri},
  title =	{{Minimum Cycle and Homology Bases of Surface Embedded Graphs}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5915},
  URN =		{urn:nbn:de:0030-drops-59152},
  doi =		{10.4230/LIPIcs.SoCG.2016.23},
  annote =	{Keywords: Cycle basis, Homology basis, Topological graph theory}
}

Keywords: Cycle basis, Homology basis, Topological graph theory
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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