Splitting (Complicated) Surfaces is Hard

Overview & Synopsis

Let $\MM$ be an orientable combinatorial surface without boundary. A cycle on $\MM$ is \emph{splitting} if it has no self-intersections and it partitions $\MM$ 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 $g^{O(g)} n \log n$ time.