### Optimal pants decompositions and shortest homotopic cycles on an orientable surfaceOptimal pants decompositions and shortest homotopic cycles on an orientable surface

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 Author Verdire, ric Colin De ♦ Lazarus, Francis Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2007 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Combinatorial optimization ♦ Combinatorial surface ♦ Computational topology ♦ Homotopy ♦ Pants decomposition Abstract 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 $\textit{optimal}:$ 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 article are polynomial, assuming uniformity of the weights of the vertex-edge graph of the surface. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2007-07-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 54 Issue Number 4

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Source: ACM Digital Library