Automorphisms and Isomorphisms of Maps in Linear Time

Authors Ken-ichi Kawarabayashi, Bojan Mohar, Roman Nedela, Peter Zeman

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Author Details

Ken-ichi Kawarabayashi
  • National Institute of Informatics, Tokyo, Japan
Bojan Mohar
  • Department of Mathematics, Simon Fraser University, Burnaby, Canada
  • IMFM, Department of Mathematics, University of Ljubljana, Slovenia
Roman Nedela
  • Univeristy of West Bohemia, Pilsen, Czech Republic
Peter Zeman
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

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Ken-ichi Kawarabayashi, Bojan Mohar, Roman Nedela, and Peter Zeman. Automorphisms and Isomorphisms of Maps in Linear Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 86:1-86:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A map is a 2-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. When the underlying surface is orientable, every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map, parametrized by the genus of the underlying surface. The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • maps on surfaces
  • automorphisms
  • isomorphisms
  • algorithm


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