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 ⩽ 5 g − 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 ⌊ 3 g / 2 ⌋ − 1 ⩽ w ⩽ ⌊ 3 g / 2 ⌋ . We conjecture that in the worst case w = 2 g + o ( g ) . For this conjecture we have two candidate classes of planar graphs.