Testing Graph Isotopy on Surfaces

TL;DR

This paper presents efficient algorithms for determining whether two graph embeddings on surfaces are isotopic, with optimal linear-time solutions in fixed arrangements and a polynomial-time algorithm in the plane minus points.

Contribution

It introduces new algorithms for the graph isotopy problem on surfaces, improving computational efficiency and providing a new proof of a mathematical characterization.

Findings

Linear-time algorithm for fixed arrangements.

O(n^{3/2} log n) algorithm in the punctured plane.

Reproof of the isotopy-homotopy congruence characterization.

Abstract

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 (1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.