Nonconstructive advances in polynomial-time complexity
Information Processing Letters
Nonconstructive tools for proving polynomial-time decidability
Journal of the ACM (JACM)
On search decision and the efficiency of polynomial-time algorithms
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
Journal of Algorithms
A simpler proof of the excluded minor theorem for higher surfaces
Journal of Combinatorial Theory Series B
Upper bounds on the size of obstructions and intertwines
Journal of Combinatorial Theory Series B
On computing graph minor obstruction sets
Theoretical Computer Science
Constructive Linear Time Algorithms for Branchwidth
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
On the excluded minors for the matroids of branch-width k
Journal of Combinatorial Theory Series B
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Obtaining a planar graph by vertex deletion
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
Parameterized Complexity
Graph and map isomorphism and all polyhedral embeddings in linear time
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Faster parameterized algorithms for minor containment
Theoretical Computer Science
Faster parameterized algorithms for minor containment
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
A parameterized complexity tutorial
LATA'12 Proceedings of the 6th international conference on Language and Automata Theory and Applications
A basic parameterized complexity primer
The Multivariate Algorithmic Revolution and Beyond
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
Effective computation of immersion obstructions for unions of graph classes
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Effective computation of immersion obstructions for unions of graph classes
Journal of Computer and System Sciences
Parameterized complexity of vertex deletion into perfect graph classes
Theoretical Computer Science
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By Robertson and Seymour's graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general method for obtaining such algorithms, which is based on definability in monadic second-order logic and the decidability of the monadic second-order theory of trees. A straightforward application of our method yields algorithms that, for a given k, compute excluded minor characterisations for the minor ideal Tk of all graphs of tree width at most k, the minor ideal Bk of all graphs of branch width at most k, and the minor ideal Gk of all graphs of genus at most k. Our main results are concerned with constructions of new minor ideals from given ones. Answering a question that goes back to Fellows and Langston [11], we prove that there is an algorithm that, given excluded minor characterisations of two minor ideals C and D, computes such a characterisation for the ideal C ∪ D. Furthermore, we obtain an algorithm for computing an excluded minor characterisation for the class of all apex graphs over a minor ideal C, given an excluded minor characterisation for C. (An apex graph over C is a graph G from which one vertex can be removed to obtain a graph in C.) A corollary of this result is a uniform ftpalgorithm for the "distance k from planarity" problem.