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ESA'05 Proceedings of the 13th annual European conference on Algorithms
Exact algorithms for the Hamiltonian cycle problem in planar graphs
Operations Research Letters
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Theoretical Computer Science
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
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Discrete Applied Mathematics
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MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
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Planar k-path in subexponential time and polynomial space
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Structure theorem and isomorphism test for graphs with excluded topological subgraphs
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Dynamic programming for graphs on surfaces
ACM Transactions on Algorithms (TALG)
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We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex H-minor-free graph G contains a path of length k in 2O(√k). nO(1) steps. Our approach builds on a combination of Demaine-Hajiaghayi's bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of H-minor-free graphs. This result is used to bound the number of ways vertex disjoint paths can be routed through the separators of such decompositions. The proof is based on several structural theorems from the Graph Minors series of Robertson and Seymour. With a slight modification, similar combinatorial and algorithmic results can be derived for many other problems. Our approach can be viewed as a general framework for obtaining time 2O(√k). nO(1) algorithms on H-minor-free graph classes.