Optimal Pants Decompositions and Shortest Homotopic Cycles on an Orientable Surface

De Verdière, Éric Colin and Lazarus, Francis (2004) Optimal Pants Decompositions and Shortest Homotopic Cycles on an Orientable Surface. In: Graph Drawing 11th International Symposium, GD 2003, September 21-24, 2003, Perugia, Italy , pp. 478-490 (Official URL: http://dx.doi.org/10.1007/978-3-540-24595-7_45).

Full text not available from this repository.


A pants decomposition of a compact orientable surface M is a set of disjoint simple cycles which cuts M into pairs of pants, i.e., spheres with three boundaries. Assuming M is a polyhedral surface, with weighted vertex-edge graph G, we consider combinatorial pants decompositions: the cycles are closed walks in G that may overlap but do not cross. We give an algorithm which, given a pants decomposition, computes a homotopic pants decomposition in which each cycle is a shortest cycle in its homotopy class. In particular, the resulting decomposition is optimal (as short as possible among all homotopic pants decompositions), and any optimal pants decomposition is made of shortest homotopic cycles. Our algorithm is polynomial in the complexity of the input and in the longest-to-shortest edge ratio of G. The same algorithm can be applied, given a simple cycle C, to compute a shortest cycle homotopic to C which is itself simple.

Item Type:Conference Paper
Additional Information:10.1007/978-3-540-24595-7_45
Classifications:G Algorithms and Complexity > G.070 Area / Edge Length
G Algorithms and Complexity > G.560 Geometry
ID Code:477

Repository Staff Only: item control page


P. Buser. Geometry and spectra of compact Riemann surfaces, volume 106 of Progress in Mathematics. Birhäuser, 1992.

É. Colin de Verdière and F. Lazarus. Optimal system of loops on an orientable surface. In IEEE Symp. Found. Comput. Sci., pages 627-636, 2002.

D. Epstein. Curves on 2-manifolds and isotopies. Acta Mathematica, 115:83-107, 1966.

J. Erickson and S. Har-Peled. Optimally cutting a surface into a disk. In Proc. 18th Annu. ACM Symp. Comput. Geom., pages 244-253, 2002.

A. Hatcher. Pants decompositions of surfaces. http://www.math.cornell.edu/~hatcher/Papers/pantsdecomp.pdf, 2000.

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications, 4:63-98, 1994.

W. S. Massey. Algebraic Topology: An Introduction, volume 56 of Graduate Texts in Mathematics. Springer-Verlag, 1977.