## Inserting an Edge Into a Planar Graph (2000)

Venue: | Algorithmica |

Citations: | 17 - 8 self |

### BibTeX

@INPROCEEDINGS{Gutwenger00insertingan,

author = {Carsten Gutwenger and Petra Mutzel and René Weiskircher},

title = {Inserting an Edge Into a Planar Graph},

booktitle = {Algorithmica},

year = {2000},

pages = {246--255},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, which is able to find a crossing minimum solution.

### Citations

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(Show Context)
Citation Context ... is among the most challenging problems in graph theory and graph drawing. Although, there is a vast amount of literature on this NP-hard problem (for a survey see, e.g., [13], NPhardness is shown in =-=[5]-=-), so far no practically efficient exact algorithm for crossing minimization is known. Currently, the best known approach for crossing minimization is based on planarization. Here, in a first step, th... |

130 |
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(Show Context)
Citation Context ...0, 4], it is often hard to optimize over the set of all possible combinatorial erabeddings. E.g., the problem of bend minimization can be solved in polynomial time for a fixed combinatorial embedding =-=[14]-=-, while it is NP-hard over the set of all combinatorial embeddings [6]. When a linear function of polynomial size is defined on the cycles of a graph, it is NP-hard to find the embedding that maximize... |

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82 | On the computational complexity of upward and rectilinear planarity testing
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(Show Context)
Citation Context ...atorial erabeddings. E.g., the problem of bend minimization can be solved in polynomial time for a fixed combinatorial embedding [14], while it is NP-hard over the set of all combinatorial embeddings =-=[6]-=-. When a linear function of polynomial size is defined on the cycles of a graph, it is NP-hard to find the embedding that maximizes the value of the cycles that are face cycles in the embedding [12, t... |

78 | Incremental planarity testing
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(Show Context)
Citation Context ...or biconnected graphs. A connected graph is biconnected if it contains no vertex whose removal splits the graph into two or more components. SPQR-trees have been suggested by Di Battista and Tamassia =-=[3]-=-. They represent a decomposition of a biconnected graph into triconnected components. A connected graph is triconnected, if there is no pair of vertices in the graph whose removal splits the graph int... |

56 | An experimental comparison of four graph drawing algorithms
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(Show Context)
Citation Context ...n of crossing minimization using planarization and a linear time implementation of SPQR-trees [7]. In our tests, we use the 8249 non-planar graphs from a benchmark set collected by Di Battista et al. =-=[2]-=- ranging from 10 to 100 vertices. The planar subgraph is computed using the AGD implementation of [9]. In 10.1 (831 graphs) of our benchmark graphs the planar subgraphs resulted from deleting a single... |

35 |
A linear time implementation of SPQRtrees
- Gutwenger, Mutzel
- 2001
(Show Context)
Citation Context ...nected planar graph. Another important property of these trees is that their size (including the skeletons) is linear in the size of the original graph and that they can be constructed in linear time =-=[7]-=-. As described in [3], SPQR-trees can be used to represent all combinatorial erabeddings of a biconnected planar graph. This is done by choosing erabeddings for the skeletons of the nodes in the tree.... |

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10 |
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- 1994
(Show Context)
Citation Context ...duction Crossing minimization is among the most challenging problems in graph theory and graph drawing. Although, there is a vast amount of literature on this NP-hard problem (for a survey see, e.g., =-=[13]-=-, NPhardness is shown in [5]), so far no practically efficient exact algorithm for crossing minimization is known. Currently, the best known approach for crossing minimization is based on planarizatio... |

4 | On computing a maximal planar subgraph using PQ-trees
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(Show Context)
Citation Context ... our tests, we use the 8249 non-planar graphs from a benchmark set collected by Di Battista et al. [2] ranging from 10 to 100 vertices. The planar subgraph is computed using the AGD implementation of =-=[9]-=-. In 10.1 (831 graphs) of our benchmark graphs the planar subgraphs resulted from deleting a single edge. In these instances, our algorithm is direct applicable. In the remaining cases, we insert the ... |

1 |
Randomized graph drawing with heavy duty preprocessing
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(Show Context)
Citation Context ...an be inserted, and so on. One criticism of the planarization method was that when choosing a "bad" embedding in the edge reinsertion phase, the number of crossings may get much higher than =-=necessary [8]-=-. Hence, the question arose if there is a polynomial time algorithm for inserting an edge into the planar subgraph P so that the number of crossings is minimized. Thereby, the task is to optimize over... |

1 | Optimizing over all combinatorial erabeddings of a planar graph - Mutzel, Weiskircher - 1999 |

1 | Computing optimal erabeddings for planar graphs - Mutzel, Weiskircher - 2000 |