# Structure theorem and isomorphism test for graphs with excluded topological subgraphs

Grohe, M and Marx, Dániel
(2012)
*Structure theorem and isomorphism test for graphs with excluded topological subgraphs.*
In: 44th Annual ACM Symposium on Theory of Computing, STOC '12, 2012-05-19 - 2012-05-22, New York, Amerikai Egyesült Államok.
10.1145/2213977.2213996

## Abstract

We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter |H|. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding H as a topological subgraph, Partial Dominating Set (find k vertices whose closed neighborhood has maximum size) can be solved in time f(H,k)·n O(1) time. More significantly, we show that on graphs excluding H as a topological subgraph, Graph Isomorphism can be solved in time n f(H). This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and H-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition. © 2012 ACM.

Item Type: | Conference or Workshop Item (-) |
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Additional Information: | #Könyv Szerző ismeretlen |

Uncontrolled Keywords: | PARAMETERS, TOPOLOGY, Trees (mathematics), Theorem proving, Set theory, Graphic methods, Forestry, Algorithms, Tree-like decomposition, Tree decomposition, Topological-minor, Subgraphs, Solvable case, Polynomial-time, Free graphs, Fixed surfaces, Fixed graphs, Dominating sets, Bounded degree, Algorithmic applications, topological minors, Graph isomorphism, Fixed-parameter tractability |

Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Depositing User: | EPrints Admin |

Date Deposited: | 16 Jan 2014 10:30 |

Last Modified: | 05 Feb 2014 12:27 |

URI: | https://eprints.sztaki.hu/id/eprint/7371 |

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