Topologically Trivial Closed Walks in Directed Surface Graphs

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 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(g 2 L 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.