Sciweavers

LICS
2010
IEEE

Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors

13 years 15 days ago
Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors
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...
Martin Grohe
Added 14 Feb 2011
Updated 14 Feb 2011
Type Journal
Year 2010
Where LICS
Authors Martin Grohe
Comments (0)