1 research outputs found
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