https://hal-lirmm.ccsd.cnrs.fr/lirmm-00904498Dorn, FredericFredericDornUiB - Department of Informatics [Bergen] - UiB - University of BergenFomin, Fedor V.Fedor V.FominUiB - Department of Informatics [Bergen] - UiB - University of BergenThilikos, Dimitrios M.Dimitrios M.ThilikosDepartment of Mathematics [Athens] - NKUA - National and Kapodistrian University of AthensALGCO - Algorithmes, Graphes et Combinatoire - LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier - UM - Université de Montpellier - CNRS - Centre National de la Recherche ScientifiqueCatalan structures and dynamic programming in H-minor-free graphsHAL CCSD2012Parameterized complexityLongest pathMinor-free graphsCatalan structure[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Thilikos, Dimitrios2013-11-14 15:43:352022-09-06 17:00:302013-11-29 15:55:37enJournal articles10.1016/j.jcss.2012.02.0041We 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 $2^(O(\sqrt{k})) n^{O(1)}$ steps. Our approach builds on a combination of Demaine-Hajiaghayiʼs 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 $2^(O(\sqrt{k})) n^{O(1)}$ algorithms on H-minor-free graph classes.