# Hanani-Tutte for approximating maps of graphs

## TL;DR

This paper extends the Hanani-Tutte theorem to graph maps on surfaces, providing a constructive proof and an efficient algorithm to determine if such maps can be approximated by embeddings.

## Contribution

It generalizes the Hanani-Tutte theorem to surface embeddings and introduces a constructive proof along with an efficient testing algorithm.

## Key findings

- Resolved conjectures on graph map approximations on surfaces.
- Provided a constructive proof leading to an efficient embedding test.
- Developed an algorithm for embedding approximation based on the theorem.

## Abstract

We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing if a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise non-intersecting "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.