Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G 1 and G 2 of the same abstract graph G on an orientable surface S , decide whether G 1 and G 2 are isotopic; in other words, whether there exists a continuous family of embeddings between G 1 and G 2 . We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G 1 (resp., G 2 ) with a fixed graph cellularly embedded on S ; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G 1 and G 2 are piecewise-linear embeddings in the plane, minus a finite set of points; our algorithm runs in O ( n 3/2 log 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 (Topology 23:303–311, 1984 ): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.