Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

For graph G, let bw(G) denote the branchwidth of G and gm(G) the largest integer g such that G contains a g×g grid as a minor. We show that bw(G) ≤ 3gm(G) + 1 for every planar graph G. This is an improvement over the bound bw(G) ≤ 4gm(G) due to Robertson, Seymour and Thomas. Our proof is constructive and implies quadratic time constant-factor approximation algorithms for planar graphs for both problems of finding a largest grid minor and of finding an optimal branch-decomposition: (3 + e)-approximation for the former and (2 + e)-approximation for the latter, where e is an arbitrary positive constant. We also study the tightness of the above bound. A k×h cylinder, denoted by C k,h , is the Cartesian product of a cycle on k vertices and a path on h vertices. We show that bw(C 2h, h ) = 2gm(C 2h,h ) = 2h and therefore the coefficient in the above upper bound is within a factor of 3/2 from the best possible.