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LICS

2010

IEEE

2010

IEEE

We prove that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. That is, for every class C of graphs such that some graph H is not a minor of any graph in C, a property P of graphs in C is decidable in polynomial time if and only if it is definable in fixed-point logic with counting. Furthermore, we prove that for every class C of graphs with excluded minors there is a k such that the k-dimensional Weisfeiler-Lehman algorithm decides isomorphism of graphs in C in polynomial time. The Weisfeiler-Lehman algorithm is a combinatorial algorithm for testing isomorphism. It generalises the basic colour refinement algorithm and is much simpler than the known group-theoretic algorithms for deciding isomorphism of graphs with excluded minors. The main technical theorem behind these two results is a "definable structure theorem" for classes of graphs with excluded minors. It states that graphs with excluded minors can be decomposed i...

Related Content

Added |
14 Feb 2011 |

Updated |
14 Feb 2011 |

Type |
Journal |

Year |
2010 |

Where |
LICS |

Authors |
Martin Grohe |

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