## Non-Planar Core Reduction of Graphs

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Citations: | 3 - 3 self |

### BibTeX

@MISC{Gutwenger_non-planarcore,

author = {Carsten Gutwenger and Markus Chimani},

title = {Non-Planar Core Reduction of Graphs},

year = {}

}

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### Abstract

We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs.

### Citations

226 | Tarjan, Efficient planarity testing
- Hopcroft, E
- 1974
(Show Context)
Citation Context ...m 1 shows a procedure for computing the non-planar core. We achieve linear running time, since constructing an SPQR-tree, testing planarity, and computing traversing costs takes only linear time; see =-=[6, 9, 8]-=-. 4 Crossing Number In this section, we apply the non-planar core reduction to the crossing number problem. The following theorem shows that it is sufficient to compute the crossing number of the non-... |

208 |
Crossing number is NP-complete
- MR, DS
- 1983
(Show Context)
Citation Context ...s which is the minimum number of planar subgraphs of G whose union is G. However, finding an optimal drawing with respect to any of these non-planarity measures yields an NP-hard optimization problem =-=[5, 12, 13]-=-. Various heuristic and exact methods for solving these optimization problems have been proposed; please refer to [11, 14, 7] for an overview. It is well known that it is sufficient to consider each b... |

165 |
Sur le problème des courbes gauches en topologie, Fund
- Kuratowski
(Show Context)
Citation Context ...ings is in general a primary objective. Such a drawing is called a planar drawing. However, it is well known that not every graph can be drawn without edge crossings. The famous theorem by Kuratowski =-=[10]-=- shows that a graph is planar if and only if it does not contain a subdivision of K3,3 or K5. If a graph G is not planar, a question arises naturally: How far away is the graph from planarity? For tha... |

78 | Incremental planarity testing
- Battista, Tamassia
- 1989
(Show Context)
Citation Context ...empty or contains both s and t. 2.3 SPQR-Trees SPQR-trees basically represent the decomposition of a biconnected graph into its triconnected components. For a formal definition we refer the reader to =-=[4, 3]-=-. Informally speaking, the nodes of an SPQR-tree T of a graph G stand for serial (S-nodes), parallel (P-nodes), and triconnected (R-nodes) structures, as well as edges of G (Q-nodes). The respective s... |

56 | An experimental comparison of four graph drawing algorithms
- Battista, Garg, et al.
- 1997
(Show Context)
Citation Context ...nesses of the non-planar R-nodes is not correct. 8 Experimental Results and Discussion We tested the effect of our reduction strategy on a widely used benchmark set commonly known as the Rome library =-=[2]-=-. This library contains over 11.000 graphs ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in real-life software engineering and database applications. We... |

35 |
A linear time implementation of SPQRtrees
- Gutwenger, Mutzel
- 2001
(Show Context)
Citation Context ...m 1 shows a procedure for computing the non-planar core. We achieve linear running time, since constructing an SPQR-tree, testing planarity, and computing traversing costs takes only linear time; see =-=[6, 9, 8]-=-. 4 Crossing Number In this section, we apply the non-planar core reduction to the crossing number problem. The following theorem shows that it is sufficient to compute the crossing number of the non-... |

32 | Planarizing graphs - a survey and annotated bibliography
- Liebers
(Show Context)
Citation Context ...any of these non-planarity measures yields an NP-hard optimization problem [5, 12, 13]. Various heuristic and exact methods for solving these optimization problems have been proposed; please refer to =-=[11, 14, 7]-=- for an overview. It is well known that it is sufficient to consider each biconnected component of the graph separately. We present a new approach based on the triconnectivity structure of the graph w... |

29 |
On the deletion of nonplanar edges of a graph
- Liu, Geldmacher
- 1977
(Show Context)
Citation Context ...s which is the minimum number of planar subgraphs of G whose union is G. However, finding an optimal drawing with respect to any of these non-planarity measures yields an NP-hard optimization problem =-=[5, 12, 13]-=-. Various heuristic and exact methods for solving these optimization problems have been proposed; please refer to [11, 14, 7] for an overview. It is well known that it is sufficient to consider each b... |

24 |
Determining the thickness of a graph is NP-hard
- Mansfield
- 1983
(Show Context)
Citation Context ...s which is the minimum number of planar subgraphs of G whose union is G. However, finding an optimal drawing with respect to any of these non-planarity measures yields an NP-hard optimization problem =-=[5, 12, 13]-=-. Various heuristic and exact methods for solving these optimization problems have been proposed; please refer to [11, 14, 7] for an overview. It is well known that it is sufficient to consider each b... |

18 | Thethickness of graphs: a survey
- Mutzel, Odenthal, et al.
- 1998
(Show Context)
Citation Context ...any of these non-planarity measures yields an NP-hard optimization problem [5, 12, 13]. Various heuristic and exact methods for solving these optimization problems have been proposed; please refer to =-=[11, 14, 7]-=- for an overview. It is well known that it is sufficient to consider each biconnected component of the graph separately. We present a new approach based on the triconnectivity structure of the graph w... |

17 | Inserting an edge into a planar graph
- Gutwenger, Mutzel, et al.
- 2005
(Show Context)
Citation Context ... connects the two faces adjacent to (s, t) without using the dual edge of (s, t). We also call the corresponding list of primal edges a traversing path for s and t. Gutwenger, Mutzel, and Weiskircher =-=[8]-=- showed that the traversing costs are independent of the choice of the embedding Γ of G. Hence, we define the traversing costs of G with respect to (s, t) to be the traversing costs of an arbitrary em... |

14 |
An experimental study of crossing minimization heuristics
- Gutwenger, Mutzel
- 2004
(Show Context)
Citation Context ...any of these non-planarity measures yields an NP-hard optimization problem [5, 12, 13]. Various heuristic and exact methods for solving these optimization problems have been proposed; please refer to =-=[11, 14, 7]-=- for an overview. It is well known that it is sufficient to consider each biconnected component of the graph separately. We present a new approach based on the triconnectivity structure of the graph w... |

3 |
On-line maintanance of triconnected components with SPQR-trees
- Battista, Tamassia
- 1996
(Show Context)
Citation Context ...empty or contains both s and t. 2.3 SPQR-Trees SPQR-trees basically represent the decomposition of a biconnected graph into its triconnected components. For a formal definition we refer the reader to =-=[4, 3]-=-. Informally speaking, the nodes of an SPQR-tree T of a graph G stand for serial (S-nodes), parallel (P-nodes), and triconnected (R-nodes) structures, as well as edges of G (Q-nodes). The respective s... |

2 |
Additivity of the crossing number of graphs with connectivity 2
- unknown authors
- 1984
(Show Context)
Citation Context ... us to restrict the crossings in which the edges of a planar st-component may be involved so that we can still obtain a crossing minimal drawing of G. A similar result has been reported by ˇ Siráň in =-=[15]-=-. However, as pointed out in [1], the proof given by ˇ Siráň is not correct. Lemma 3. Let C = (VC, EC) be a planar st-component of G = (V, E). Then, there exists a crossing minimal drawing D∗ of G suc... |

1 |
On the minimum cut of planarizations
- Chimani, Gutwenger
- 2005
(Show Context)
Citation Context ...which the edges of a planar st-component may be involved so that we can still obtain a crossing minimal drawing of G. A similar result has been reported by ˇ Siráň in [15]. However, as pointed out in =-=[1]-=-, the proof given by ˇ Siráň is not correct. Lemma 3. Let C = (VC, EC) be a planar st-component of G = (V, E). Then, there exists a crossing minimal drawing D∗ of G such that the induced drawing D∗ C ... |