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)
Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
The graph genus problem is NP-complete
Journal of Algorithms
Computational complexity of combinatorial surfaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Some provably hard crossing number problems
Discrete & Computational Geometry
Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Discrete Applied Mathematics - Special issue: Combinatorial Optimization 1992 (CO92)
Separating and nonseparating disjoint homotopic cycles in graph embeddings
Journal of Combinatorial Theory Series B
Embedding graphs in an arbitrary surface in linear time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface
SIAM Journal on Discrete Mathematics
Highly connected sets and the excluded grid theorem
Journal of Combinatorial Theory Series B
Journal of the ACM (JACM)
Depth-First Search and Kuratowski Subgraphs
Journal of the ACM (JACM)
Face covers and the genus problem for Apex graphs
Journal of Combinatorial Theory Series B
Computing a canonical polygonal schema of an orientable triangulated surface
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Computing crossing numbers in quadratic time
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Optimally cutting a surface into a disk
Proceedings of the eighteenth annual symposium on Computational geometry
Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas
SIAM Journal on Computing
Crossing numbers of random graphs
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Which Crossing Number is it, Anyway?
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
On determining the genus of a graph in O(v O(g)) steps(Preliminary Report)
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Improving the crossing lemma by finding more crossings in sparse graphs: [extended abstract]
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
European Journal of Combinatorics - Special issue: Topological graph theory
Graph Theory With Applications
Graph Theory With Applications
Parameterized Complexity
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Graph and map isomorphism and all polyhedral embeddings in linear time
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Testing contractibility in planar rips complexes
Proceedings of the twenty-fourth annual symposium on Computational geometry
Improved upper bounds on the crossing number
Proceedings of the twenty-fourth annual symposium on Computational geometry
Inserting a vertex into a planar graph
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Untangling Tanglegrams: Comparing Trees by Their Drawings
ISBRA '09 Proceedings of the 5th International Symposium on Bioinformatics Research and Applications
Minimum cuts and shortest homologous cycles
Proceedings of the twenty-fifth annual symposium on Computational geometry
A Practical Approach to Courcelle's Theorem
Electronic Notes in Theoretical Computer Science (ENTCS)
Approximating the crossing number of toroidal graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Crossing numbers and parameterized complexity
GD'07 Proceedings of the 15th international conference on Graph drawing
A shorter proof of the graph minor algorithm: the unique linkage theorem
Proceedings of the forty-second ACM symposium on Theory of computing
Proceedings of the forty-second ACM symposium on Theory of computing
Adding one edge to planar graphs makes crossing number hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Linkless and flat embeddings in 3-space and the unknot problem
Proceedings of the twenty-sixth annual symposium on Computational geometry
Output-sensitive algorithm for the edge-width of an embedded graph
Proceedings of the twenty-sixth annual symposium on Computational geometry
ACM Transactions on Algorithms (TALG)
Approximating the crossing number of graphs embeddable in any orientable surface
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Do we really understand the crossing numbers?
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Untangling Tanglegrams: Comparing Trees by Their Drawings
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
On the red/blue spanning tree problem
Theoretical Computer Science
An algorithm for the graph crossing number problem
Proceedings of the forty-third annual ACM symposium on Theory of computing
Shortest non-trivial cycles in directed surface graphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
A satisfiability-based approach for embedding generalized tanglegrams on level graphs
SAT'11 Proceedings of the 14th international conference on Theory and application of satisfiability testing
Facets in the Crossing Number Polytope
SIAM Journal on Discrete Mathematics
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Minimum cuts and shortest non-separating cycles via homology covers
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Algorithms for the edge-width of an embedded graph
Computational Geometry: Theory and Applications
FPT suspects and tough customers: open problems of downey and fellows
The Multivariate Algorithmic Revolution and Beyond
Generalized k-ary tanglegrams on level graphs: A satisfiability-based approach and its evaluation
Discrete Applied Mathematics
New races in parameterized algorithmics
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Obtaining planarity by contracting few edges
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Obtaining planarity by contracting few edges
Theoretical Computer Science
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We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph in the plane with at most k crossings. This answers the question posed by Grohe (STOC'01 and JCSS 2004). Our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan lin1, which determines if a given graph is planar in linear time. Our algorithm can also be compared to the algorithms by Mohar (STOC'96 and Siam J. Discrete Math 2001), for testing the embeddability of an input graph in a fixed surface. For each surface s, Mohar describes an algorithm which yields either an embedding of G in s or a minor of G which is not embeddable in s and is minimal with this property. The same approach allows us to obtain linear time algorithms for the same question for a variety of other crossing numbers. We can also determine in linear time if an input graph can be made planar by the deletion of k edges (for fixed k).