Optimally cutting a surface into a disk

We consider the problem of cutting a subset of the edges of 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 in general, 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(log2 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 [20]. 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.

• Type: Journal
• Year: 2002
• Where: CORR
• Authors: Jeff Erickson, Sariel Har-Peled
• Keywords: CORR 2002, Education, Minimum Cut Graph, Polyhedral Manifold Surface, Single Topological Disk