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 + 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. A k×h cylinder, denoted by Ck,h, is the Cartesian product of a cycle on k vertices and a path on h vertices. We show that bw(C2h, h) = 2gm(C2h,h) = 2h and therefore the coefficient in the above upper bound is within a factor of 3/2 from the best possible. © 2010 Springer-Verlag.
CITATION STYLE
Gu, Q. P., & Tamaki, H. (2010). Improved bounds on the planar branchwidth with respect to the largest grid minor size. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6507 LNCS, pp. 85–96). https://doi.org/10.1007/978-3-642-17514-5_8
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