On the homotopy test on surfaces

TL;DR

This paper introduces linear-time algorithms for testing homotopy of closed walks on surfaces, with applications to surface groups, using a geometric approach that leverages preprocessing for efficiency.

Contribution

It provides the first linear-time algorithms for homotopy testing of closed walks on surfaces, extending to non-orientable cases with fixed basepoints.

Findings

Homotopy test runs in O(|c|+|d|) time after O(|G|) preprocessing.

Linear time algorithms for word and conjugacy problems in surface groups.

Applicable to both orientable and non-orientable surfaces under specified conditions.

Abstract

Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed basepoint. We restrict S to be orientable for the free homotopy test, but allow non-orientable surfaces when the basepoint is fixed. After O(|G|) time preprocessing independent of c and d, our algorithms answer the homotopy test in O(|c|+|d|) time, where |G|, |c| and |d| are the respective numbers of edges of G, c and d. As a byproduct we obtain linear time algorithms for the word problem and the conjugacy problem in surface groups. We present a geometric approach based on previous works by Colin de Verdi\`ere and Erickson.