Finding $k$ Disjoint Paths in a Directed Planar Graph
SIAM Journal on Computing
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
NP-completeness of some edge-disjoint paths problems
Discrete Applied Mathematics
The equivalence of theorem proving and the interconnection problem
ACM SIGDA Newsletter
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
Lower bounds on the pathwidth of some grid-like graphs
Discrete Applied Mathematics
Algorithms for finding an induced cycle in planar graphs and bounded genus graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Graph minors. XXI. Graphs with unique linkages
Journal of Combinatorial Theory Series B
Induced Packing of Odd Cycles in a Planar Graph
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
The induced disjoint paths problem
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
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
Slightly superexponential parameterized problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Graph Minors. XXII. Irrelevant vertices in linkage problems
Journal of Combinatorial Theory Series B
Linear kernels for (connected) dominating set on H-minor-free graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Planar disjoint-paths completion
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
Kernel bounds for path and cycle problems
Theoretical Computer Science
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The Disjoint-Paths Problem asks, given a graph G and a set of pairs of terminals (s1, t1),..., (sk, tk), whether there is a collection of k pairwise vertex-disjoint paths linking si and ti, for i = 1,..., k. In their f(k) ċ n3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an "irrelevant" vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose - very technical - proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we prove this result for planar graphs achieving g(k) = 2O(k). Our bound is radically better than the bounds known for general graphs. Moreover, our proof is new and self-contained, and it strongly exploits the combinatorial properties of planar graphs. We also prove that our result is optimal, in the sense that the function g(k) cannot become better than exponential. Our results suggest that any algorithm for the DISJOINT-PATHS PROBLEM that runs in time better than 22o(k) ċnO(1) will probably require drastically different ideas from those in the irrelevant vertex technique.