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Algorithmica

, Volume 64, Issue 1, pp 69–84 | Cite as

Fast Minor Testing in Planar Graphs

  • Isolde Adler
  • Frederic DornEmail author
  • Fedor V. Fomin
  • Ignasi Sau
  • Dimitrios M. Thilikos
Article

Abstract

Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components C u and C v . A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time \(\mathcal{O}(2^{\mathcal{O}(h)} \cdot n +n^{2}\cdot\log n)\) a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.

Keywords

Graph minors Planar graphs Branchwidth Parameterized complexity Dynamic programming 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Isolde Adler
    • 1
  • Frederic Dorn
    • 2
    Email author
  • Fedor V. Fomin
    • 2
  • Ignasi Sau
    • 3
  • Dimitrios M. Thilikos
    • 4
  1. 1.Institut für InformatikGoethe-UniversitätFrankfurtGermany
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.AlGCo teamCNRS, LIRMMMontpellierFrance
  4. 4.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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