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Fast Minor Testing in Planar Graphs

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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.

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Author information

Authors and Affiliations

  1. Institut für Informatik, Goethe-Universität, Frankfurt, Germany

    Isolde Adler

  2. Department of Informatics, University of Bergen, Bergen, Norway

    Frederic Dorn & Fedor V. Fomin

  3. AlGCo team, CNRS, LIRMM, Montpellier, France

    Ignasi Sau

  4. Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece

    Dimitrios M. Thilikos

Authors
  1. Isolde Adler
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  2. Frederic Dorn
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  3. Fedor V. Fomin
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  4. Ignasi Sau
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  5. Dimitrios M. Thilikos
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Corresponding author

Correspondence to Frederic Dorn.

Additional information

An extended abstract of this work appeared in the proceedings of ESA’10 [2].

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Adler, I., Dorn, F., Fomin, F.V. et al. Fast Minor Testing in Planar Graphs. Algorithmica 64, 69–84 (2012). https://doi.org/10.1007/s00453-011-9563-9

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  • DOI: https://doi.org/10.1007/s00453-011-9563-9

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