## Abstract

The H-Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H-Minor containment is one of the most important and technical parts of the Graph Minor Theory of Robertson and Seymour and it is a cornerstone for most of the algorithmic applications of this theory. H-Minor containment for graphs of bounded branchwidth is a basic ingredient of this algorithm. The currently fastest solution to this problem, based on the ideas introduced by Robertson and Seymour, was given by Hicks in [I.V. Hicks, Branch decompositions and minor containment, Networks 43 (1) (2004) 19], providing an algorithm that in time O(3^{k2}·(h+k-1)!·m) decides if a graph G with m edges and branchwidth k, contains a fixed graph H on h vertices as a minor. In this work we improve the dependence on k of Hicks' result by showing that checking if H is a minor of G can be done in time O(2^{(2k+1)·logk}·h ^{2k}·2^{2h2}·m). We set up an approach based on a combinatorial object called rooted packing, which captures the properties of the subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first algorithm for minor containment testing with single-exponential dependence on branchwidth. Namely, it runs in time 2^{O(k)}·h^{2k}·2^{O(h)}·n, with n=|V(G)|. Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction containment.

Original language | English (US) |
---|---|

Pages (from-to) | 7018-7028 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 412 |

Issue number | 50 |

DOIs | |

State | Published - Nov 25 2011 |

## Keywords

- Branchwidth
- Dynamic programming
- Graph minor containment
- Graph minors
- Graphs on surfaces
- Parameterized complexity

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science