Automorphism groups of maps in linear time

By a map we mean 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. 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.

[1]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[2]  Andrej Brodnik,et al.  The simultaneous conjugacy problem in the symmetric group , 2019 .

[3]  Rudolf Mathon,et al.  A Note on the Graph Isomorphism counting Problem , 1979, Inf. Process. Lett..

[4]  H. Whitney.  2-Isomorphic Graphs , 1933 .

[5]  David M. Mount,et al.  Isomorphism of graphs with bounded eigenvalue multiplicity , 1982, STOC '82.

[6]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[7]  Martin Grohe,et al.  A Faster Isomorphism Test for Graphs of Small Degree , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[8]  Andrej Brodnik,et al.  Fast permutation-word multiplication and the simultaneous conjugacy problem , 2019, ArXiv.

[9]  Jonathan L. Gross,et al.  Handbook of graph theory , 2007, Discrete mathematics and its applications.

[10]  Max Fontet.  Calcul du centralisateur d'un groupe de permutations , 1977 .

[11]  Roman Nedela,et al.  Exponents of orientable maps , 1997 .

[12]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[13]  Uwe Schöning.  Graph Isomorphism is in the Low Hierarchy , 1988, J. Comput. Syst. Sci..

[14]  A. Pultr,et al.  Combinatorial, algebraic, and topological representations of groups, semigroups, and categories , 1980 .

[15]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[16]  Michal Pilipczuk,et al.  Fixed-Parameter Tractable Canonization and Isomorphism Test for Graphs of Bounded Treewidth , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[17]  G. Jones,et al.  Theory of Maps on Orientable Surfaces , 1978 .

[18]  Arthur T. White,et al.  Permutation Groups and Combinatorial Structures , 1979 .

[19]  Christoph M. Hoffmann.  Subcomplete Generalizations of Graph Isomorphism , 1982, J. Comput. Syst. Sci..

[20]  László Babai,et al.  Graph isomorphism in quasipolynomial time [extended abstract] , 2015, STOC.

[21]  Carsten Thomassen,et al.  Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface , 1991 .

[22]  S. Fisk.  Geometric coloring theory , 1977 .

[23]  Ken-ichi Kawarabayashi.  Graph Isomorphism for Bounded Genus Graphs In Linear Time , 2015, ArXiv.

[24]  László Babai,et al.  Vertex-transitive graphs and vertex-transitive maps , 1991, J. Graph Theory.

[25]  Ken-ichi Kawarabayashi,et al.  Graph and map isomorphism and all polyhedral embeddings in linear time , 2008, STOC.

[26]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[27]  P. Mani.  Automorphismen von polyedrischen Graphen , 1971 .

[28]  Amos Altshuler,et al.  Construction and enumeration of regular maps on the torus , 1973, Discret. Math..

[29]  Charles J. Colbourn,et al.  Linear Time Automorphism Algorithms for Trees, Interval Graphs, and Planar Graphs , 1981, SIAM J. Comput..

[30]  Gary L. Miller,et al.  Isomorphism testing for graphs of bounded genus , 1980, STOC '80.

[31]  Robert E. Tarjan,et al.  A V log V Algorithm for Isomorphism of Triconnected Planar Graphs , 1973, J. Comput. Syst. Sci..