## Efficient Extraction of Multiple Kuratowski Subdivisions (2007)

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Citations: | 2 - 1 self |

### BibTeX

@MISC{Chimani07efficientextraction,

author = {Markus Chimani and Petra Mutzel and Jens M. Schmidt},

title = {Efficient Extraction of Multiple Kuratowski Subdivisions },

year = {2007}

}

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### Abstract

A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for non-planarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., Branch-and-Cut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation.

### Citations

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Citation Context ... their common endpoints. Graphs which admit such a planar drawing, are called planar graphs, and recognizing this graph class has been a vivid research topic for the past decades. Hopcroft and Tarjan =-=[12]-=- showed in 1974 that this problem can be solved in linear time, using sophisticated data structures and intricate algorithms. Current planarity testing algorithms like the ones by Boyer and Myrvold [4... |

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Citation Context ...r of identified subdivisions dramatically, albeit on the cost of the running time. To speed up the backtracking subroutine, it is possible to use algorithms for dynamic connectivity for planar graphs =-=[8]-=-. This increases the overall runtime only by a factor of log(n) in comparison to the linear time approach in terms of output complexity. 3.5 Runtime Analysis As described in Section 3.2, our strategy ... |

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Citation Context ...omes the dominant term. The bundle variant uses a traditional back-tracking scheme and therefore does not guarantee the theoretical logarithmic bound. We use the graphs of the well-known Rome Library =-=[2]-=-, which contains 11528 real-world graphs with 10 to 100 nodes, 8249 of which are non-planar graphs. We also use random graphs (n = 10 . . . 500, m = 2n) generated by OGDF. Thereby we start with an emp... |

32 |
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Citation Context ...efficient extraction of such a witness of non-planarity was non-trivial in the context of the first linear planarity tests. Linear algorithms for such an extraction were later presented by Williamson =-=[17]-=- and Karabeg [14]. Modern planarity testing algorithms like the ones by Boyer and Myrvold, and de Fraysseix et al. can directly extract a single Kuratowski subdivision, if the given graph is non-plana... |

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Citation Context ...eix et al. can directly extract a single Kuratowski subdivision, if the given graph is non-planar. In ILP-based Branch-and-Cut approaches which try to solve, e.g., the Maximum Planar Subgraph problem =-=[13]-=- or the Crossing Minimization problem [6,7],s2 Markus Chimani, Petra Mutzel, and Jens M. Schmidt the identification of multiple such witnesses is a crucial part. Thereby, we look at some intermediate ... |

25 | On the cutting edge: Simplified O(n) planarity by edge addition
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Citation Context ...2] showed in 1974 that this problem can be solved in linear time, using sophisticated data structures and intricate algorithms. Current planarity testing algorithms like the ones by Boyer and Myrvold =-=[4,5]-=- and de Fraysseix, Ossona de Mendez and Rosenstiehl [9,10,11] are less complex but still quite involved. As shown by Kuratowski [15] in 1930, a graph is planar if and only if it does not contain a K3,... |

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Citation Context ...2] showed in 1974 that this problem can be solved in linear time, using sophisticated data structures and intricate algorithms. Current planarity testing algorithms like the ones by Boyer and Myrvold =-=[4,5]-=- and de Fraysseix, Ossona de Mendez and Rosenstiehl [9,10,11] are less complex but still quite involved. As shown by Kuratowski [15] in 1930, a graph is planar if and only if it does not contain a K3,... |

15 |
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Citation Context ...ar time, using sophisticated data structures and intricate algorithms. Current planarity testing algorithms like the ones by Boyer and Myrvold [4,5] and de Fraysseix, Ossona de Mendez and Rosenstiehl =-=[9,10,11]-=- are less complex but still quite involved. As shown by Kuratowski [15] in 1930, a graph is planar if and only if it does not contain a K3,3 or a K5 subdivision, i.e., a complete bipartite graph K3,3 ... |

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Citation Context ...|. This runtime is linear in the graph size and the extracted Kuratowski edges. The algorithm is based on the planarity test of Boyer and Myrvold [5] which is one of the fastest planarity tests today =-=[3]-=-. We will only give a short introduction into this planarity test in Section 2; for a full description of the original test see [5]. The main part of this paper focuses on the description on how to mo... |

10 |
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Citation Context ...ar time, using sophisticated data structures and intricate algorithms. Current planarity testing algorithms like the ones by Boyer and Myrvold [4,5] and de Fraysseix, Ossona de Mendez and Rosenstiehl =-=[9,10,11]-=- are less complex but still quite involved. As shown by Kuratowski [15] in 1930, a graph is planar if and only if it does not contain a K3,3 or a K5 subdivision, i.e., a complete bipartite graph K3,3 ... |

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8 |
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Citation Context ...uratowski subdivision, if the given graph is non-planar. In ILP-based Branch-and-Cut approaches which try to solve, e.g., the Maximum Planar Subgraph problem [13] or the Crossing Minimization problem =-=[6,7]-=-,s2 Markus Chimani, Petra Mutzel, and Jens M. Schmidt the identification of multiple such witnesses is a crucial part. Thereby, we look at some intermediate solution and try to find Kuratowski subdivi... |

4 |
the Open Graph Drawing Framework
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(Show Context)
Citation Context ...of the algorithm is O(n + m + log n � K∈S |E(K)|). 4 Experimental Analysis We implemented the algorithm and its bundle variant as part of the open-source C++-based Open Graph Drawing Framework (OGDF) =-=[1]-=-. All tests were performed on an Intel Core2Duo E6300 with 1.86 GHz and 2GB RAM using the GNU-compiler gcc-3.4.4 (-o1). Due to the algorithmic complexities, we simplified the steps to compute the crit... |

4 |
Classification and detection of obstructions to planarity
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(Show Context)
Citation Context ...ion of such a witness of non-planarity was non-trivial in the context of the first linear planarity tests. Linear algorithms for such an extraction were later presented by Williamson [17] and Karabeg =-=[14]-=-. Modern planarity testing algorithms like the ones by Boyer and Myrvold, and de Fraysseix et al. can directly extract a single Kuratowski subdivision, if the given graph is non-planar. In ILP-based B... |

4 |
le probleme des corbes gauches en topologie. Fundamenta Mathematicae 15
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(Show Context)
Citation Context ...nt planarity testing algorithms like the ones by Boyer and Myrvold [4,5] and de Fraysseix, Ossona de Mendez and Rosenstiehl [9,10,11] are less complex but still quite involved. As shown by Kuratowski =-=[15]-=- in 1930, a graph is planar if and only if it does not contain a K3,3 or a K5 subdivision, i.e., a complete bipartite graph K3,3 or complete graph K5 with edges replaced by paths of length at least on... |

2 |
Fraysseix and P. Ossona de Mendez. On cotree-critical and DFS cotree-critical graphs
- de
(Show Context)
Citation Context ...ar time, using sophisticated data structures and intricate algorithms. Current planarity testing algorithms like the ones by Boyer and Myrvold [4,5] and de Fraysseix, Ossona de Mendez and Rosenstiehl =-=[9,10,11]-=- are less complex but still quite involved. As shown by Kuratowski [15] in 1930, a graph is planar if and only if it does not contain a K3,3 or a K5 subdivision, i.e., a complete bipartite graph K3,3 ... |

2 |
Effiziente Extraktion von Kuratowski-Teilgraphen
- Schmidt
- 2007
(Show Context)
Citation Context ...S |E(K)|). We will only give a brief sketch of the proof, ands14 Markus Chimani, Petra Mutzel, and Jens M. Schmidt omit a number of rather technical case differentiations. For a detailed analysis see =-=[16]-=-. For the minor-types E1–E5 and AE1–AE4 it is necessary to compute at least one external node strictly below the existing highest-xy-path in the stopping configuration. Such nodes are called external ... |