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

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

0.00 Avg rating0 Votes
Article ID: iaor20125370
Volume: 64
Issue: 3
Start Page Number: 416
End Page Number: 453
Publication Date: Nov 2012
Journal: Algorithmica
Authors: ,
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)≤3 gm(G) for every planar graph G. This is an improvement over the bound bw(G)≤4 gm(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 b w ( G ) c g m ( G ) + o ( g m ( G ) ) equ1 does not hold in general for a planar graph G.

Reviews

Required fields are marked *. Your email address will not be published.