On the homotopy test on surfaces

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.

💡 Research Summary

The paper addresses the fundamental computational problem of testing whether two closed walks c and d in a graph G that is cellularly embedded on a surface S are homotopic. Two variants are considered: (i) homotopy with a fixed basepoint and (ii) free homotopy, where the basepoint may move. The authors work in the RAM model and achieve linear‑time performance both for preprocessing the embedding and for answering each homotopy query.

The main contributions are as follows. First, after an O(|G|) preprocessing phase that is independent of the walks, the algorithm can decide basepoint‑fixed homotopy in O(|c| + |d|) time. The preprocessing builds a minimum cut graph that turns the surface into a topological polygon, extracts a system of 2g (or 2h for non‑orientable) fundamental loops, and stores for every edge its crossing information with respect to these loops. Each walk is then scanned once; each traversed edge contributes a signed count to the corresponding loop, producing an integer vector that uniquely represents the homotopy class of the walk relative to the chosen basepoint. Equality of the two vectors is both necessary and sufficient for basepoint‑fixed homotopy.

Second, for free homotopy the authors restrict to orientable surfaces (the problem becomes NP‑hard on non‑orientable ones). Using the same cut‑graph structure, each walk is converted into a word over the generators of the surface’s fundamental group. The words are reduced by cancelling inverse pairs, and a linear‑time string‑matching routine (e.g., Knuth‑Morris‑Pratt) checks whether the reduced words are cyclic shifts of one another. If they are, the walks are freely homotopic; otherwise they are not. This yields an O(|c| + |d|) decision procedure after the same O(|G|) preprocessing.

A notable by‑product is that the same framework solves the word problem and the conjugacy problem in surface groups in linear time. The word problem reduces to testing whether two walks represent the same reduced word, while the conjugacy problem reduces to testing whether the reduced words are cyclically equivalent. Both checks are performed in O(|c| + |d|) time.

The technical core builds on earlier work by Colin de Verdière and Erickson on cut graphs and canonical systems of loops. The authors refine those constructions to guarantee that the crossing data can be accessed in constant time per edge, and they separate preprocessing from query handling so that the latter depends only on the lengths of the input walks. The correctness proofs rely on classical algebraic topology: the cut graph yields a presentation of π₁(S), the integer vectors (or reduced words) are invariants under homotopy, and cyclic equivalence captures the freedom of moving the basepoint along the surface.

Complexity analysis shows that the preprocessing uses O(|G|) time and space, while each query uses O(|c| + |d|) time and O(1) additional space. The algorithms are deterministic, simple to implement, and suitable for practical applications such as real‑time robot motion planning on surfaces, mesh processing, and computational group theory.

In conclusion, the paper delivers the first linear‑time algorithms for both fixed‑basepoint and free homotopy testing on embedded graphs, together with linear‑time solutions to the word and conjugacy problems in surface groups. The work bridges combinatorial topology and algorithm design, opening avenues for extensions to dynamic embeddings, higher‑dimensional manifolds, and non‑orientable free‑homotopy testing.