Automorphism groups of maps in linear time

TL;DR

This paper introduces a linear-time algorithm for computing the automorphism groups of maps embedded on surfaces, applicable to both orientable and non-orientable cases, significantly improving efficiency over previous methods.

Contribution

It presents the first linear-time algorithm for automorphism group computation of maps on surfaces, using local reductions and a uniform map approach.

Findings

Algorithm runs in linear time relative to map size

Automorphism groups can be reconstructed efficiently from uniform maps

Extension to non-orientable surfaces via antipodal double-cover

Abstract

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.