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

Given a graph GG with tree-width ω(G)ω(G), branch-width β(G)β(G), and side size of the largest square grid-minor θ(G)θ(G), it is known that θ(G)≤β(G)≤ω(G)+1≤32β(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 θ(G)=β(G)<ω(G)=⌊32θ(G)⌋−1.