Toroidal grid minors and stretch in embedded graphs

TL;DR

This paper explores the relationship between the size of the largest toroidal grid minor in an embedded graph, a new embedding density parameter called stretch, and the planar crossing number, providing bounds and approximation algorithms.

Contribution

Introduces the stretch parameter for embedded graphs and establishes bounds on toroidal expanse related to genus and degree, linking it to crossing number.

Findings

Bounds on toroidal expanse using stretch parameter

Relation between stretch, toroidal minors, and crossing number

Efficient approximation algorithms for toroidal expanse and crossing number

Abstract

We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding density parameter, the stretch of an embedded graph G, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar crossing number of G. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse and for the crossing number of a surface-embedded graph with bounded maximum degree.