A note on planar graphs with large width parameters and small grid-minors

Given a graph gg with tree-width ?(g)?(g), branch-width ß(g)ß(g), and side size of the largest square grid-minor ?(g)?(g), it is known that ?(g)=ß(g)=?(g)+1=32ß(g)?(g)=ß(g)=?(g)+1=32ß(g). In this paper, we introduce another approach to bound the side size of the largest square grid-minor specifically for planar graphs. The approach is based on measuring the distances between the faces in an embedding of a planar graph. We analyze the tightness of all derived bounds. In particular, we present a class of planar graphs where ?(g)=ß(g)?(g)=ß(g)?(g)=?32?(g)?-1