Computing minimum length paths of a given homotopy class
Computational Geometry: Theory and Applications
Making curves minimally crossing by Reidemeister moves
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Transforming curves on surfaces
Journal of Computer and System Sciences - Special issue on the 36th IEEE symposium on the foundations of computer science
Using generic programming for designing a data structure for polyhedral surfaces
Computational Geometry: Theory and Applications - Special issue on applications and challenges
Dynamic generators of topologically embedded graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Computing homotopic shortest paths in the plane
Journal of Algorithms
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal pants decompositions and shortest homotopic cycles on an orientable surface
Journal of the ACM (JACM)
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Computing homotopic shortest paths efficiently
Computational Geometry: Theory and Applications
Tightening Nonsimple Paths and Cycles on Surfaces
SIAM Journal on Computing
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We investigate the following problem: Given two embeddings G1 and G2 of the same abstract graph G on an orientable surface S, decide whether G1 and G2 are isotopic; in other words, whether there exists a continuous family of embeddings between G1 and G2. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G1 (resp., G2) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G1 and G2 are piecewise-linear embeddings in the plane minus a finite set of points; our algorithm runs in O(n3/2log n) time, where n is the complexity of the input. The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine. As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.