The "minor crossing number" of a graph G is the minimum crossing number of a graph that contains G as a minor. It is proved that for every graph H there is a constant c, such that every graph G with no H-minor has minor crossing number at most c∣V(G)∣.
@article{arxiv.math/0609707,
title = {The Minor Crossing Number of Graphs with an Excluded Minor},
author = {Drago Bokal and Gašper Fijavž and David R. Wood},
journal= {arXiv preprint arXiv:math/0609707},
year = {2008}
}