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JACM
2007

Optimal pants decompositions and shortest homotopic cycles on an orientable surface

8 years 3 months ago
Optimal pants decompositions and shortest homotopic cycles on an orientable surface
We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertex-edge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal: each cycle is as short as possible in its homotopy class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this paper are polynomial, assuming uniformity of ...
Éric Colin de Verdière, Francis Laza
Added 15 Dec 2010
Updated 15 Dec 2010
Type Journal
Year 2007
Where JACM
Authors Éric Colin de Verdière, Francis Lazarus
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