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 $$g\ge 2$$ . 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.

中文翻译：

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有向曲面图中的拓扑平凡封闭游走

设 G 是一个有 n 个顶点和 m 个边的有向图，嵌入在表面 S 上，可能有边界，第一个 Betti 数为 $$\beta $$ 。我们考虑了在 G 中寻找可收缩（同伦无关）或有界（整数同调无关）的封闭有向游走的复杂性。具体来说，我们描述了确定 G 是否包含 $O( n+m)$$ 时间，或在 $$O(n+m)$$ 时间内的收缩封闭行走，或在 $$O(\beta(n+m))$$ 时间内的有界封闭行走。我们的算法依赖于 G 和对偶图 G* 中强连通性之间的微妙关系；我们的可收缩封闭游走算法也依赖于 Hass 和 Scott 的开创性拓扑结果。我们还证明了检测简单的边界环是 NP-hard 的。我们还描述了三种多项式时间算法来计算最短的可收缩闭合游走，这取决于表面的基本群是自由的、阿贝尔的还是双曲线的。我们的双曲曲面算法的一个关键步骤是构建一个具有 $$O(g^2L^2)$$ 非终结符的上下文无关文法，该文法生成所有长度最多为 L 的可收缩闭合游走，并且仅可收缩闭合在 $$g\ge 2$$ 属的四边形系统中行走。最后，我们表明计算最短的简单可收缩循环、最短的简单有界循环和最短的有界封闭游走都是 NP-hard。最后，当输入图的某些边具有负权重时，我们考虑检测具有平凡拓扑的负封闭步行的复杂性。我们表明可以在多项式时间内检测到负边界游走，通过减少到最大流量。我们还描述了多项式时间算法，以在环面或任何具有边界的表面上的图形中找到负可收缩游走；双曲曲面的其余情况保持开放。简单循环的相应问题都是 NP-hard。