research-article

Graph and map isomorphism and all polyhedral embeddings in linear time

Authors:

Ken-ichi Kawarabayashi,

Bojan MoharAuthors Info & Claims

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

Pages 471 - 480

https://doi.org/10.1145/1374376.1374443

Published: 17 May 2008 Publication History

Abstract

For every surface S (orientable or non-orientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of face-width at least 3 into S. This improves a previously known algorithm whose time complexity is n^O(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k ≥ 3, we want to find an embedding of G in S of face-width at least k, or conclude that such an embedding does not exist. It is known that this problem is NP-hard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of face-width at least k, up to Whitney equivalence. Here, the face-width of an embedded graph G is the minimum number of points of G in which some non-contractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3-connected graphs. The second ingredient is a linear time algorithm for map isomorphism and Whitney equivalence. This part generalizes the seminal result of Hopcroft and Wong that graph isomorphism can be decided in linear time for planar graphs.

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Cited By

Kawarabayashi KMohar BNedela RZeman P(2024)Automorphisms and Isomorphisms of Maps in Linear TimeACM Transactions on Algorithms10.1145/368679821:1(1-32)Online publication date: 22-Nov-2024
https://dl.acm.org/doi/10.1145/3686798
Golovach PStamoulis GThilikos D(2023)Hitting Topological Minor Models in Planar Graphs is Fixed Parameter TractableACM Transactions on Algorithms10.1145/358368819:3(1-29)Online publication date: 10-Feb-2023
https://dl.acm.org/doi/10.1145/3583688
Soto Sánchez JMedeiros E Sá AFigueiredo L(2019)Acquiring periodic tilings of regular polygons from imagesThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-019-01665-y35:6-8(899-907)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s00371-019-01665-y
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Index Terms

Graph and map isomorphism and all polyhedral embeddings in linear time
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory

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cover image ACM Conferences

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

May 2008

712 pages

ISBN:9781605580470

DOI:10.1145/1374376

General Chair:
Richard Ladner
University of Washington
,
Program Chair:
Cynthia Dwork
Microsoft Research, Silicon Valley

Copyright © 2008 ACM.

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Published: 17 May 2008

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Cited By

Kawarabayashi KMohar BNedela RZeman P(2024)Automorphisms and Isomorphisms of Maps in Linear TimeACM Transactions on Algorithms10.1145/368679821:1(1-32)Online publication date: 22-Nov-2024
https://dl.acm.org/doi/10.1145/3686798
Golovach PStamoulis GThilikos D(2023)Hitting Topological Minor Models in Planar Graphs is Fixed Parameter TractableACM Transactions on Algorithms10.1145/358368819:3(1-29)Online publication date: 10-Feb-2023
https://dl.acm.org/doi/10.1145/3583688
Soto Sánchez JMedeiros E Sá AFigueiredo L(2019)Acquiring periodic tilings of regular polygons from imagesThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-019-01665-y35:6-8(899-907)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s00371-019-01665-y
Lokshtanov DRamanuajn MSaurabh SCzumaj A(2018)When recursion is better than iterationProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175430(1916-1933)Online publication date: 7-Jan-2018
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Lokshtanov DRamanujan MSaurabh S(2018)Linear Time Parameterized Algorithms for Subset Feedback Vertex SetACM Transactions on Algorithms10.1145/315529914:1(1-37)Online publication date: 3-Jan-2018
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Lokshtanov DRamanujan MSaurabh S(2015)Linear Time Parameterized Algorithms for Subset Feedback Vertex SetAutomata, Languages, and Programming10.1007/978-3-662-47672-7_76(935-946)Online publication date: 20-Jun-2015
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Lokshtanov DPilipczuk MPilipczuk MSaurabh S(2014)Fixed-Parameter Tractable Canonization and Isomorphism Test for Graphs of Bounded TreewidthProceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2014.28(186-195)Online publication date: 18-Oct-2014
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Fox KKhanna S(2013)Shortest non-trivial cycles in directed and undirected surface graphsProceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms10.5555/2627817.2627843(352-364)Online publication date: 6-Jan-2013
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Kawarabayashi KKreutzer SMohar B(2012)Linkless and Flat Embeddings in 3-SpaceDiscrete & Computational Geometry10.5555/3116278.311656447:4(731-755)Online publication date: 1-Jun-2012
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Cabello SColin de Verdière íLazarus F(2012)Algorithms for the edge-width of an embedded graphComputational Geometry: Theory and Applications10.1016/j.comgeo.2011.12.00245:5-6(215-224)Online publication date: 1-Jul-2012
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