A GAP package for computation with coherent configurations
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Fixed-point definability and polynomial time on chordal graphs and line graphs
Fields of logic and computation
Sherali-Adams relaxations and indistinguishability in counting logics
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Experimental descriptive complexity
Logic and Program Semantics
Fixed-point definability and polynomial time on graphs with excluded minors
Journal of the ACM (JACM)
On tractable parameterizations of graph isomorphism
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Practical graph isomorphism, II
Journal of Symbolic Computation
Maximum Matching and Linear Programming in Fixed-Point Logic with Counting
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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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-Leman algorithm decides isomorphism of graphs in C in polynomial time. The Weisfeiler-Leman 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 "definables tructure theorem" for classes of graphs with excluded minors. It states that graphs with excluded minors can be decomposed into pieces arranged in a treelike structure, together with a linear order of each of the pieces. Furthermore, the decomposition and the linear orders on the pieces are definable in fixed-point logic (without counting).