Given a simple planar graph with tree-width w and side size of the largest square grid minor g, it is known that g?w?5g-1g?w?5g-1. Thus, the side size of the largest grid minor is a constant approximation for the tree-width in planar graphs. In this work we analyze the lower bounds of this approximation. In particular, we present a class of planar graphs with ?3g/2?-1?w?3g/2??3g/2?-1?w??3g/2?. We conjecture that in the worst case w=2g+o(g)w=2g+o(g). For this conjecture we have two candidate classes of planar graphs.
Grigoriev, A., Marchal, L., & Usotskaya, N. (2009). On planar graphs with large tree-width and small grid minors. Electronic Notes in Discrete Mathematics, 32, 35-42. https://doi.org/10.1016/j.endm.2009.02.006