Topologically Trivial Closed Walks in Directed Surface Graphs

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. Finally, we consider the complexity of detecting negative closed walks with trivial topology, when some edges of the input graph have negative weights. We show that negative bounding walks can be detected in polynomial time, by reduction to maximum flows. We also describe polynomial-time algorithms to find negative contractible walks in graphs on the torus or any surface with boundary; the remaining case of hyperbolic surfaces remains open. The corresponding problems for simple cycles are all NP-hard.