## Elimination of local bridges (1997)

Venue: | Math. Slovaca |

Citations: | 7 - 7 self |

### BibTeX

@ARTICLE{Juvan97eliminationof,

author = {Martin Juvan and Bojan Mohar},

title = {Elimination of local bridges},

journal = {Math. Slovaca},

year = {1997},

volume = {47},

pages = {85--92}

}

### OpenURL

### Abstract

Let K be a subgraph of G. It is shown that if G is 3–connected modulo K then it is possible to replace branches of K by other branches joining same pairs of main vertices of K such that G has no local bridges with respect to the new subgraph K. A linear time algorithm is presented that either performs such a task, or finds a Kuratowski subgraph K5 or K3,3 in a subgraph of G formed by a branch e and local bridges on e. This result is needed in linear time algorithms for embedding graphs in surfaces.

### Citations

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(Show Context)
Citation Context ...determine whether the graph is planar or not. The first such algorithm was obtained by Hopcroft and Tarjan [6] back in 1974. There are several other linear time planarity algorithms (Booth and Lueker =-=[1]-=-, Fraysseix and Rosenstiehl [4], Williamson [13, 14]). The extensions of original algorithms produce also an embedding (rotation system) whenever the given graph is found to be planar [2], or find a s... |

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Citation Context .... We believe that our results can also be used in some other problems involving bridges (see, e.g., [12]). In our algorithm, we need plane embeddings of graphs. These can be described combinatorially =-=[5]-=- by specifying a rotation system: for each vertex v of the graph G we have the cyclic permutation πv of its neighbours, representing their circular order around v on the surface. In order to make a cl... |

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Citation Context ...t of a larger project [8, 9] which shows that there is a linear time algorithm to construct embeddings of graphs in an arbitrary (fixed) surface, generalizing the well–known Hopcroft–Tarjan algorithm =-=[6]-=- testing planarity in linear time. These algorithms rely on the theory of bridges: a subgraph K of G is embedded in the surface and then this embedding is either extended to an embedding of G, or an o... |

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Citation Context ...h and Lueker [1], Fraysseix and Rosenstiehl [4], Williamson [13, 14]). The extensions of original algorithms produce also an embedding (rotation system) whenever the given graph is found to be planar =-=[2]-=-, or find a small obstruction — a subgraph homeomorphic to K5 or K3,3 — if the graph is non-planar [13, 14]. Concerning the time complexity of our algorithms, we assume a randomaccess machine (RAM) mo... |

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Citation Context ...determine whether the graph is planar or not. The first such algorithm was obtained by Hopcroft and Tarjan [6] back in 1974. There are several other linear time planarity algorithms (Booth and Lueker =-=[1, 8]-=-, Fraysseix and Rosenstiehl [4], Williamson [14, 15]). Extensions of original algorithms produce also an embedding (described by a rotation system) whenever the given graph is found to be planar [2], ... |

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Citation Context ... non-planar [13, 14]. Concerning the time complexity of our algorithms, we assume a randomaccess machine (RAM) model with unit cost for basic operations. This model was introduced by Cook and Reckhow =-=[3]-=-. More precisely, our model is the unit-cost RAM where operations on integers, whose value is O(n), need only constant time (n is the size of the given graph). 2 Elimination of local bridges In many c... |

50 | A linear time algorithm for embedding graphs in an arbitrary surface
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(Show Context)
Citation Context ...a homeomorphic subgraph K 0 having the same set of main vertices such that no K 0 -bridge in G is local. Our main motivation comes from considering algorithmic aspects of embedding extension problems =-=[7, 10]-=-. Algorithms developed in [7, 10] rely on the theory of bridges: a subgraph K of G is embedded in the surface and then this embedding is either extended to an embedding of G, or an obstruction for suc... |

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Citation Context ...he first such algorithm was obtained by Hopcroft and Tarjan [6] back in 1974. There are several other linear time planarity algorithms (Booth and Lueker [1], Fraysseix and Rosenstiehl [4], Williamson =-=[13, 14]-=-). The extensions of original algorithms produce also an embedding (rotation system) whenever the given graph is found to be planar [2], or find a small obstruction — a subgraph homeomorphic to K5 or ... |

20 |
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Citation Context ...planar or not. The first such algorithm was obtained by Hopcroft and Tarjan [6] back in 1974. There are several other linear time planarity algorithms (Booth and Lueker [1], Fraysseix and Rosenstiehl =-=[4]-=-, Williamson [13, 14]). The extensions of original algorithms produce also an embedding (rotation system) whenever the given graph is found to be planar [2], or find a small obstruction — a subgraph h... |

18 |
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Citation Context ...he first such algorithm was obtained by Hopcroft and Tarjan [6] back in 1974. There are several other linear time planarity algorithms (Booth and Lueker [1], Fraysseix and Rosenstiehl [4], Williamson =-=[13, 14]-=-). The extensions of original algorithms produce also an embedding (rotation system) whenever the given graph is found to be planar [2], or find a small obstruction — a subgraph homeomorphic to K5 or ... |

13 |
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Citation Context ...y selecting the “other” facial walk W ′ of the face containing xy on its boundary. Let e ′ := W ′ − xy be the new branch replacing e ′ . One can show that e ′ has no local bridges attached to it (see =-=[11]-=- for details). 5sOtherwise, let L be a Kuratowski subgraph from the planarity test for N + xy. Note that L ′ can be obtained in linear time by the algorithm of Williamson [13, 14]. It is clear that L ... |

8 |
Cycles and bridges in graphs
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Citation Context ...ficulties in this approach are local bridges. In this paper it is shown how to overcome this problem. We believe that our results can also be used in some other problems involving bridges (see, e.g., =-=[12]-=-). In our algorithm, we need plane embeddings of graphs. These can be described combinatorially [5] by specifying a rotation system: for each vertex v of the graph G we have the cyclic permutation πv ... |

8 |
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Citation Context ...ions is found. One of the difficulties in achieving linear time complexity is the presence of local bridges. Elimination of local bridges is a useful tool also in disjoint paths problems (cf. Ohtsuki =-=[11]-=-, Robertson and Seymour [12]). Similar application is in graph drawing [9]. We believe that our results can also be used in some other problems involving bridges (see, e.g., [13]). In our algorithm, w... |

4 |
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(Show Context)
Citation Context ...K, then G + (K) is 3-connected. On the other hand, if K is homeomorphic to a 3-connected graph, then G is 3-connected modulo K if and only if it is 3-connected. This paper is part of a larger project =-=[8, 9]-=- which shows that there is a linear time algorithm to construct embeddings of graphs in an arbitrary (fixed) surface, generalizing the well–known Hopcroft–Tarjan algorithm [6] testing planarity in lin... |

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3 |
An outline of a disjoint paths algorithm, in: "Paths, Flows, and VLSI-Layout
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(Show Context)
Citation Context ...fficulties in achieving linear time complexity is the presence of local bridges. Elimination of local bridges is a useful tool also in disjoint paths problems (cf. Ohtsuki [11], Robertson and Seymour =-=[12]-=-). Similar application is in graph drawing [9]. We believe that our results can also be used in some other problems involving bridges (see, e.g., [13]). In our algorithm, we need plane embeddings of g... |

2 |
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(Show Context)
Citation Context ...K, then G + (K) is 3-connected. On the other hand, if K is homeomorphic to a 3-connected graph, then G is 3-connected modulo K if and only if it is 3-connected. This paper is part of a larger project =-=[8, 9]-=- which shows that there is a linear time algorithm to construct embeddings of graphs in an arbitrary (fixed) surface, generalizing the well–known Hopcroft–Tarjan algorithm [6] testing planarity in lin... |

1 |
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(Show Context)
Citation Context ...e proof of Proposition 2.1 yields a quadratic time algorithm for the local bridges elimination. It is possible to improve it into an O(n log n) algorithm by some additional more sophisticated methods =-=[7]-=-. However, in various applications (e.g., [8, 9]), a linear time procedure is desired. A solution that is suitable for the applications in surface embedding algorithms is presented in this section. If... |

1 |
Convex representations of maps on the torus and other flat surfaces, Discrete Comput. Geom
- Mohar
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(Show Context)
Citation Context ...is the presence of local bridges. Elimination of local bridges is a useful tool also in disjoint paths problems (cf. Ohtsuki [11], Robertson and Seymour [12]). Similar application is in graph drawing =-=[9]-=-. We believe that our results can also be used in some other problems involving bridges (see, e.g., [13]). In our algorithm, we need plane embeddings of graphs. These can be described combinatorially ... |