On planar graphs with large tree-width and small grid minors

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−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⌋. We conjecture that in the worst case w=2g+o(g). For this conjecture we have two candidate classes of planar graphs.