Splitting (complicated) surfaces is hard

Authors:
Erin W. Chambers;Éric Colin de Verdière;Jeff Erickson;Francis Lazarus;Kim Whittlesey
Affiliations:
University of Illinois;CNRS, ècole normale supérieure, Paris, France;University of Illinois;CNRS, Laboratoire des Images et des Signaux, Grenoble, France;University of Illinois
Venue:
Proceedings of the twenty-second annual symposium on Computational geometry
Year:
2006

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Abstract

Let M be an orientable combinatorial surface without boundary. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither of which is homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in gO(g)n log n time.