Toroidal Grid Minors and Stretch in Embedded Graphs∗

1 We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest 2 toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding 3 density parameter, the stretch of an embedded graph G, and use it to bound the toroidal 4 expanse from above and from below within a constant factor depending only on the genus and 5 the maximum degree. We also show that these parameters are tightly related to the planar 6 crossing number of G. As a consequence of our bounds, we derive an efficient constant factor 7 approximation algorithm for the toroidal expanse and for the crossing number of a surface8 embedded graph with bounded maximum degree. 9