CFI Construction and Balanced Graphs
FAW '09 Proceedings of the 3d International Workshop on Frontiers in Algorithmics
From polynomial time queries to graph structure theory
Proceedings of the 13th International Conference on Database Theory
Fixed-point definability and polynomial time
CSL'09/EACSL'09 Proceedings of the 23rd CSL international conference and 18th EACSL Annual conference on Computer science logic
Fixed-point definability and polynomial time on chordal graphs and line graphs
Fields of logic and computation
Structure theorem and isomorphism test for graphs with excluded topological subgraphs
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Where First-Order and Monadic Second-Order Logic Coincide
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Fixed-point definability and polynomial time on graphs with excluded minors
Journal of the ACM (JACM)
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We introduce a notion of definable tree decompositions of graphs. Actually, adefinable tree decomposition of a graph is not just a tree decomposition, but a more complicated structure that represents many different tree decompositions of the graph. It is definable in the graph by a tuple of formulas of some logic. In this paper, only study tree decomposition definable in fixed-point logic. We say that a definable tree decomposition is over a class of graphs if the pieces of the decomposition are in this class. We prove two general theorems lifting definability results from the pieces of a tree decomposition of a graph to the whole graph. Besides unifying earlier work on fixed-point definability and descriptive complexity theory on planar graphs and graphs of bounded tree width, these general results can be used to prove that the class of all graphs without a K_5-minor is definable infixed-point logic and that fixed-point logic with counting captures polynomialtime on this class.