Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality

  • Authors:

  • Erik D. Demaine;MohammadTaghi Hajiaghayi

  • Affiliations:

  • MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA;MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA

  • Venue:
  • SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2005

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Abstract

We prove that any H-minor-free graph, for a fixed graph H, of treewidth ω has an Ω(ω) × Ω(ω) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial con-sequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.