2 research outputs found

    Topologically Trivial Closed Walks in Directed Surface Graphs

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    Let GG be a directed graph with nn vertices and mm edges, embedded on a surface SS, possibly with boundary, with first Betti number β\beta. We consider the complexity of finding closed directed walks in GG that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in SS. Specifically, we describe algorithms to determine whether GG contains a simple contractible cycle in O(n+m)O(n+m) time, or a contractible closed walk in O(n+m)O(n+m) time, or a bounding closed walk in O(β(n+m))O(\beta (n+m)) time. Our algorithms rely on subtle relationships between strong connectivity in GG and in the dual graph G∗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)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≥2g\ge2. Finally, we show that computing shortest simple contractible cycles, shortest simple bounding cycles, and shortest bounding closed walks are all NP-hard.Comment: 30 pages, 18 figures; fixed several minor bugs and added one figure. An extended abstraction of this paper will appear at SOCG 201
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