2 Optimally Cutting a Surface into a Disk ∗

We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time nO(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log g)-approximation of the minimum cut graph in O(g2n log n) time. A preliminary version of this paper was presented at the 18th Annual ACM Symposium on Computational Geometry [19]. See http://www.cs.uiuc.edu/∼jeffe/pubs/schema.html for the most recent version of this paper. Partially supported by a Sloan Fellowship, NSF CAREER award CCR-0093348, and NSF ITR grant DMR-0121695. Partially supported by NSF CAREER award CCR-0132901. Optimally Cutting a Surface into a Disk 1