Graph minors. V. Excluding a planar graph
Journal of Combinatorial Theory Series B
Nonconstructive tools for proving polynomial-time decidability
Journal of the ACM (JACM)
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Finding approximate separators and computing tree width quickly
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Approximating treewidth, pathwidth, frontsize, and shortest elimination tree
Journal of Algorithms
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs
European Journal of Combinatorics
Highly connected sets and the excluded grid theorem
Journal of Combinatorial Theory Series B
The extremal function for complete minors
Journal of Combinatorial Theory Series B
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
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)
Bidimensional Parameters and Local Treewidth
SIAM Journal on Discrete Mathematics
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Approximation Schemes for First-Order Definable Optimisation Problems
LICS '06 Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science
The complexity of homomorphism and constraint satisfaction problems seen from the other side
Journal of the ACM (JACM)
Quickly deciding minor-closed parameters in general graphs
European Journal of Combinatorics
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Improved Approximation Algorithms for Minimum Weight Vertex Separators
SIAM Journal on Computing
On tree width, bramble size, and expansion
Journal of Combinatorial Theory Series B
A constructive proof of the Lovász local lemma
Proceedings of the forty-first annual ACM symposium on Theory of computing
MFCS '09 Proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science 2009
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
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
Treewidth lower bounds with brambles
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Parameterized Complexity
Improved algorithm for the half-disjoint paths problem
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Algorithmic meta-theorems for restrictions of treewidth
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Breaking o(n1/2)-approximation algorithms for the edge-disjoint paths problem with congestion two
Proceedings of the forty-third annual ACM symposium on Theory of computing
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
Large-treewidth graph decompositions and applications
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Lower bounds on the complexity of MSO1 model-checking
Journal of Computer and System Sciences
Hi-index | 0.00 |
Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter; the only other result with a similar flavor that is known to us is due to Demaine and Hajiaghayi and obtains a doubly-exponential bound on the parameter (albeit, for a more general class of parameterized problems). The second part of our work deals with providing a lower bound to Courcelle's famous theorem from almost two decades ago, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Whereas much work has been done on designing, improving, and applying algorithms on graphs of bounded treewidth, not much is known on the side of lower bounds: what bound on the treewidth of a class of graphs "forbids" polynomial-time parameterized algorithms to decide MSO-sentences? This question has only recently received attention with the first systematic study appearing in [Kreutzer 2009]. Using our results from the first part of our work we can improve on it significantly and establish a strong lower bound for Courcelle's theorem on classes of colored graphs.