Toroidal grid minors and stretch in embedded graphs

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. © 2019 Elsevier Inc.

Compact surfaces; Crossing number; Edge-width; Graph embeddings; Stretch; Toroidal grid

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