Improved bounds on the planar branchwidth with respect to the largest grid minor size

Abstract

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) 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 + ε)-approximation for the former and (2 + ε)-approximation for the latter, where ε is an arbitrary positive constant. We also study the tightness of the above bound. We show that for any constant c < 2, the bound of bw(G) ≤ c gm(G)+ o(gm(G)) does not hold in general for a planar graph G. © 2012 Springer Science+Business Media, LLC.