# Optimizing the Graph Minors Weak Structure Theorem

3 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : One of the major results of [N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B, 63 (1995), pp. 65--110], also known as the weak structure theorem, reveals the local structure of graphs excluding some graph as a minor: each such graph $G$ either has small treewidth or contains the subdivision of a planar graph (a wall) that can be arranged in a flat manner inside $G$, given that some small set of vertices is removed. We prove an optimized version of that theorem where (i) the relation between the treewidth of the graph and the height of the wall is linear (thus best possible) and (ii) the number of vertices to be removed is minimized.
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Journal articles

https://hal-lirmm.ccsd.cnrs.fr/lirmm-00904527
Contributor : Dimitrios Thilikos <>
Submitted on : Thursday, November 14, 2013 - 4:07:57 PM
Last modification on : Tuesday, June 30, 2020 - 3:34:02 PM

### Citation

Archontia C. Giannopoulou, Dimitrios M. Thilikos. Optimizing the Graph Minors Weak Structure Theorem. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2013, 27 (3), pp.1209-1227. ⟨10.1137/110857027⟩. ⟨lirmm-00904527⟩

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