Topologically trivial closed walks in directed surface graphs

With Yipu Wang*.

🔥 Discrete & Computational Geometry 64(4):1253–1294, 2020.
(Special issue of invited papers from the 35th International Symposium on Computational Geometry)

Proceedings of the 35th International Symposium on Computational Geometry, 34:1–34:17, 2019.


Let G be a directed graph with n vertices and m edges, embedded on a surface S, possibly with boundary, with first Betti number β. We consider the complexity of finding closed directed walks in G that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in S. Specifically, we describe algorithms to determine whether G contains a simple contractible cycle in O(n+m) time, or a contractible closed walk in O(n+m) time, or a bounding closed walk in O(β(n+m)) time. Our algorithms rely on subtle relationships between strong connectivity in G and in the dual graph G*; our contractible-closed-walk algorithm also relies on a seminal topological result of Hass and Scott. We also prove that detecting simple bounding cycles is NP-hard.

We also describe three polynomial-time algorithms to compute shortest contractible closed walks, depending on whether the fundamental group of the surface is free, abelian, or hyperbolic. A key step in our algorithm for hyperbolic surfaces is the construction of a context-free grammar with O(g2L2) non-terminals that generates all contractible closed walks of length at most L, and only contractible closed walks, in a system of quads of genus g≥2. Finally, we show that computing shortest simple contractible cycles, shortest simple bounding cycles, and shortest bounding closed walks are all NP-hard.

Publications - Jeff Erickson ( 07 Dec 2020