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

Authors:
Alexander Grigoriev;Bert Marchal;Natalya Usotskaya;Ioan Todinca
Affiliations:
Maastricht University SBE, Maastricht, The Netherlands;Maastricht University SBE, Maastricht, The Netherlands;Maastricht University SBE, Maastricht, The Netherlands;LIFO, Université d'Orleans, Orleans, France
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

Given a graph G with tree-width @w(G), branch-width @b(G), and side size of the largest square grid-minor @q(G), it is known that @q(G)@?@b(G)@?@w(G)+1@?32@b(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 @q(G)=@b(G)