Topologically Trivial Closed Walks in Directed Surface Graphs
Jeff Erickson, Yipu Wang
TL;DR
This paper develops efficient algorithms for detecting and computing contractible and bounding closed walks in directed surface graphs, revealing complexity results and polynomial-time solutions for specific surface types.
Contribution
It introduces new algorithms for identifying and computing topologically trivial closed walks in directed surface graphs, with complexity analyses and NP-hardness proofs.
Findings
Algorithms for simple contractible cycle detection in O(n+m) time
Algorithms for contractible closed walks in O(n+m) time
NP-hardness of detecting simple bounding cycles
Abstract
Let be a directed graph with vertices and edges, embedded on a surface , possibly with boundary, with first Betti number . We consider the complexity of finding closed directed walks in that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in . Specifically, we describe algorithms to determine whether contains a simple contractible cycle in time, or a contractible closed walk in time, or a bounding closed walk in time. Our algorithms rely on subtle relationships between strong connectivity in and in the dual graph ; 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…
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Taxonomy
TopicsAlgorithms and Data Compression · semigroups and automata theory · Advanced Combinatorial Mathematics