## Practical Toroidality Testing (1996)

Venue: | Proc. of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms |

Citations: | 10 - 2 self |

### BibTeX

@INPROCEEDINGS{Neufeld96practicaltoroidality,

author = {Eugene Neufeld and Wendy Myrvold},

title = {Practical Toroidality Testing},

booktitle = {Proc. of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms},

year = {1996},

pages = {574--580}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the graphs on up to 10 vertices and the 11 vertex ones up to 24 edges, and of these 3884 are topological obstructions, and 2249 are also minor order obstructions. A cursory search of 12 and 13 vertex graphs resulted in several more. We purport that this approach has proved practical as it has permitted us to compile what we believe to be the biggest collection of torus obstructions in the world to date. 1 Introduction A graph is said to be embedded on a surface if it is drawn there with no crossing edges. A graph is planar if it can be drawn on the sphere, and is toroidal if it can be drawn on the torus (a sphere with one handle). The genus of a planar graph is zero, and a non-planar graph which ...

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Citation Context ...hat is happening. Complete details and combinatorial proofs can be found in [16]. There have been many algorithms amenable to implementation for determining if a graph is planar including for example =-=[4, 13, 1]-=-, and also some which operate in linear time [9, 22, 2]. Parallel algorithms have also been considered with more recent work including [12, 17]. In contrast, algorithms for toroidality [8, 7, 5, 19, 2... |

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Citation Context ...oofs can be found in [16]. There have been many algorithms amenable to implementation for determining if a graph is planar including for example [4, 13, 1], and also some which operate in linear time =-=[9, 22, 2]-=-. Parallel algorithms have also been considered with more recent work including [12, 17]. In contrast, algorithms for toroidality [8, 7, 5, 19, 20, 15] have mainly been presented for their theoretical... |

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Citation Context ...rder obstructions to toroidality is finite. So far, we have succeeded in enumerating only a partial list of these. We applied Brendan McKay's makeg program based on his graph isomorphism tester nauty =-=[14]-=- to generate isomorphically distinct small graphs. The graphs tested were blocks with no degree two vertices since all obstructions which are not a block are composed of blocks which are one of the tw... |

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Citation Context ...oofs can be found in [16]. There have been many algorithms amenable to implementation for determining if a graph is planar including for example [4, 13, 1], and also some which operate in linear time =-=[9, 22, 2]-=-. Parallel algorithms have also been considered with more recent work including [12, 17]. In contrast, algorithms for toroidality [8, 7, 5, 19, 20, 15] have mainly been presented for their theoretical... |

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Citation Context ...mple [4, 13, 1], and also some which operate in linear time [9, 22, 2]. Parallel algorithms have also been considered with more recent work including [12, 17]. In contrast, algorithms for toroidality =-=[8, 7, 5, 19, 20, 15]-=- have mainly been presented for their theoretical interest, and to date, we believe that none have been implemented. The first which could lead to a nice program is the newly announced work of Juvan, ... |

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Citation Context ...determining if a graph is planar including for example [4, 13, 1], and also some which operate in linear time [9, 22, 2]. Parallel algorithms have also been considered with more recent work including =-=[12, 17]-=-. In contrast, algorithms for toroidality [8, 7, 5, 19, 20, 15] have mainly been presented for their theoretical interest, and to date, we believe that none have been implemented. The first which coul... |

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Citation Context ...oofs can be found in [16]. There have been many algorithms amenable to implementation for determining if a graph is planar including for example [4, 13, 1], and also some which operate in linear time =-=[9, 22, 2]-=-. Parallel algorithms have also been considered with more recent work including [12, 17]. In contrast, algorithms for toroidality [8, 7, 5, 19, 20, 15] have mainly been presented for their theoretical... |

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Citation Context ...hat the contraction of any edge results in a graph with lower genus, then we say that G is also a minor order obstruction. It was conjectured by Wagner in [21], and proved by Robertson and Seymour in =-=[18]-=-, that the number of minor order obstructions to toroidality is finite. So far, we have succeeded in enumerating only a partial list of these. We applied Brendan McKay's makeg program based on his gra... |

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Citation Context ...l obstruction G also satisfies the property that the contraction of any edge results in a graph with lower genus, then we say that G is also a minor order obstruction. It was conjectured by Wagner in =-=[21]-=-, and proved by Robertson and Seymour in [18], that the number of minor order obstructions to toroidality is finite. So far, we have succeeded in enumerating only a partial list of these. We applied B... |

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Citation Context ...d for their theoretical interest, and to date, we believe that none have been implemented. The first which could lead to a nice program is the newly announced work of Juvan, Marincek, and Mohar (JMM) =-=[11]-=-. Their algorithm is O(n) but a O(n 2 ) variant should prove easier Supported by NSERC. to implement. Quite likely, a program arising from their work will not be available until at least one or two ye... |

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Citation Context ...ty are harder to verify, but we have some evidence the program is behaving correctly. Enquiries about such lists generated by others have been fruitless (although a few examples from Decker's thesis, =-=[3]-=-, have been checked, and a list of cubic irreducible graphs of genus one is rumoured to exist). Uncertainty about the computed results however, is inevitable, and perhaps the greatest value of the lis... |

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