Discrete & Computational Geometry
(Special issue of invited papers from the 18th Annual ACM Symposium on Computational Geometry)
Proceedings of the 18th Annual ACM Symposium on Computational Geometry, 244-253, 2002.
We consider the problem of cutting a subset of edges of a triangulated oriented 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 number of vertices, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs an O(log² g)-approximation of the minimum cut graph in O(g²n log n) time.
The algorithms in Section 5 do not correctly compute the shortest essential cycle in surfaces with boundary, although they do correctly compute the shortest non-contractible cycle in any surface. (A cycle is contractible if it is homotopic to a point; a cycle is inessential homotopic to a point or a boundary cycle.) See “Computing the shortest essential cycle” for an explanation of the error and correct algorithms.