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Topologically Trivial Closed Walks in Directed Surface Graphs

Authors Jeff Erickson , Yipu Wang

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Author Details

Jeff Erickson
  • University of Illinois at Urbana-Champaign, Urbana, IL, USA
Yipu Wang
  • University of Illinois at Urbana-Champaign, Urbana, IL, USA

Funding

Research on this paper was partially supported by NSF grant CCF-1408763.

Acknowledgements

The first author would like to thank Tillmann Miltzow for asking an annoying question that led to this work, and to apologize for still being unable to answer it.

Cite As Get BibTex

Jeff Erickson and Yipu Wang. Topologically Trivial Closed Walks in Directed Surface Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.34

Abstract

Let G be a directed graph with n vertices and m edges, embedded on a surface S, possibly with boundary, with first Betti number beta. 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(beta (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(g^2L^2) 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • computational topology
  • surface-embedded graphs
  • homotopy
  • homology
  • strong connectivity
  • hyperbolic geometry
  • medial axes
  • context-free grammars

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