Embeddings of graphs with no short noncontractible cycles
Journal of Combinatorial Theory Series B
Easy problems for tree-decomposable graphs
Journal of Algorithms
The monadic second-order logic of graphs VII: graphs as relational structures
Theoretical Computer Science - Special issue on logic and applications to computer science
Monadic second-order evaluations on tree-decomposable graphs
Theoretical Computer Science - Special issue on selected papers of the International Workshop on Computing by Graph Transformation, Bordeaux, France, March 21–23, 1991
Journal of Combinatorial Theory Series B
Journal of the ACM (JACM)
Locality of order-invariant first-order formulas
ACM Transactions on Computational Logic (TOCL)
Deciding first-order properties of locally tree-decomposable structures
Journal of the ACM (JACM)
Fixed-Parameter Tractability, Definability, and Model-Checking
SIAM Journal on Computing
Query evaluation via tree-decompositions
Journal of the ACM (JACM)
Treewidth: Algorithmoc Techniques and Results
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Computing crossing numbers in quadratic time
Journal of Computer and System Sciences - STOC 2001
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
On Locality and Uniform Reduction
LICS '05 Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
Finding Branch-Decompositions and Rank-Decompositions
SIAM Journal on Computing
On the parameterised intractability of monadic second-order logic
CSL'09/EACSL'09 Proceedings of the 23rd CSL international conference and 18th EACSL Annual conference on Computer science logic
Lower Bounds for the Complexity of Monadic Second-Order Logic
LICS '10 Proceedings of the 2010 25th Annual IEEE Symposium on Logic in Computer Science
Deciding First-Order Properties for Sparse Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Treewidth: characterizations, applications, and computations
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Regular tree languages definable in FO
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
First-Order and Monadic Second-Order Model-Checking on Ordered Structures
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Complete graph minors and the graph minor structure theorem
Journal of Combinatorial Theory Series B
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Model checking problems for first- and monadic second-order logic on graphs have received considerable attention in the past, not the least due to their connections to problems in algorithmic graph structure theory. While the model checking problem for these logics on general graphs is computationally intractable, it becomes tractable on important classes of graphs such as those of bounded tree-width, planar graphs or more generally, classes of graphs excluding a fixed minor. It is well known that allowing an order relation or successor function can greatly increase the expressive power of the respective logics. This remains true even in cases where we require the formulas to be order- or successor-invariant, that is, while they can use an order relation, their truth in a given graph must not depend on the particular ordering or successor function chosen. Naturally, the question arises whether this increase in expressive power comes at a cost in terms of tractability on specific classes of graphs. In LICS 2012, Engel Mann et al. studied this problem and showed that order-invariant monadic second-order logic (MSO) remains tractable on the same classes of graphs than MSO without an ordering. That is, adding order-invariance to MSO essentially comes at no extra cost in terms of model checking complexity. For successor-invariant first-order logic something similar should be true. However, they only managed to show that successor-invariant first-order logic is tractable on the class of planar graphs which is very far from the best tractability results currently known for first-order logic. In this paper we significantly improve the latter result and show that successor-invariant first-order logic is tractable on any class of graphs excluding a fixed minor. This is much closer to the best results known for FO without an ordering. The proof relies on the construction of k-walks in suitable super graphs of the input graphs, i.e., walks which visit every vertex at least once and at most k times, for some k depending on the excluded minor H. The super graphs may in general contain H minors, but they still exclude some possible larger minorH0, so by results of Flum and Grohe [20] model checking on these graphs is still fixed-parameter tractable.