## Embedding graphs containing K5-subdivisions

Venue: | Ars Combinatoria |

Citations: | 8 - 2 self |

### BibTeX

@ARTICLE{Gagarin_embeddinggraphs,

author = {Andrei Gagarin and William Kocay},

title = {Embedding graphs containing K5-subdivisions},

journal = {Ars Combinatoria},

year = {},

volume = {64},

pages = {2002}

}

### OpenURL

### Abstract

Given a non-planar graph G with a subdivision of K5 as a subgraph, we can either transform the K5-subdivision into a K3,3-subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3-subdivision in the graph G. It significantly simplifies algorithms presented in [7], [10] and [12]. We then need to consider only the embeddings on the given surface of a K3,3-subdivision, which are much less numerous than those of K5. 1.

### Citations

1264 |
Graph Theory with Applications
- Bondy, Murty
- 1976
(Show Context)
Citation Context ...o consider only the embeddings on the given surface of a K3,3-subdivision, which are much less numerous than those of K5. 1. Introduction We use basic graph-theoretic terminology from Bondy and Murty =-=[1]-=- and Diestel [2]. Let G be a 2-connected, undirected, simple graph. We are interested in a practical efficient algorithm to decide whether G can be embedded in the projective plane or torus. Known alg... |

310 |
Topological Graph Theory
- Gross, Tucker
- 1987
(Show Context)
Citation Context ...he result is a considerable simplification of the existing algorithms. One needs a combinatorial description of a graph embedded on a surface. Such a description is provided by a rotation system (cf. =-=[5]-=-) of the * This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. 1sgraph, which is a set of cyclically ordered adjacency lists of its vert... |

226 | Tarjan, Efficient planarity testing
- Hopcroft, E
- 1974
(Show Context)
Citation Context ...e obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple planarity checks =-=[6]-=- or to a K3,3-subdivision in the graph G. It significantly simplifies algorithms presented in [7], [10] and [12]. We then need to consider only the embeddings on the given surface of a K3,3-subdivisio... |

165 |
Sur le problème des courbes gauches en topologie, Fund
- Kuratowski
(Show Context)
Citation Context ... an edge is +1 or −1. It is negative when the edge goes ”over the boundary” and positive otherwise. For a more detailed description see [5]. The following theorem is well known. Kuratowski’s Theorem. =-=[9]-=- A graph G is non-planar if and only if it contains a subdivision of K3,3 or K5. Hopcroft and Tarjan [6] developed an efficient practical linear time algorithm to check if a graph G is planar or not. ... |

69 |
Graph Theory, 2nd Edition
- Diestel
- 2000
(Show Context)
Citation Context ...the embeddings on the given surface of a K3,3-subdivision, which are much less numerous than those of K5. 1. Introduction We use basic graph-theoretic terminology from Bondy and Murty [1] and Diestel =-=[2]-=-. Let G be a 2-connected, undirected, simple graph. We are interested in a practical efficient algorithm to decide whether G can be embedded in the projective plane or torus. Known algorithms in [7], ... |

50 | A linear time algorithm for embedding graphs in an arbitrary surface - Mohar - 1999 |

13 |
Projective planarity in linear time
- Mohar
- 1993
(Show Context)
Citation Context ...ojective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3-subdivision in the graph G. It significantly simplifies algorithms presented in [7], =-=[10]-=- and [12]. We then need to consider only the embeddings on the given surface of a K3,3-subdivision, which are much less numerous than those of K5. 1. Introduction We use basic graph-theoretic terminol... |

10 | Practical toroidality testing - Myrvold, Neufeld - 1997 |

6 |
Searching for K3,3 in linear time
- Fellows, Kaschube
- 1991
(Show Context)
Citation Context ...the K5-subdivision (see Fig. 4). We begin by proving some basic structural results for graphs containing a T K5. Similar structural results have been proved previously by M. Fellows 2sand P. Kaschube =-=[3]-=-. We note that their proof of Theorem 1 [3] is missing the case indicated by Fig. 1 of Proposition 2.1. 2.1 Proposition. [3] A non-planar graph G with a K5-subdivision T K5 for which there is either a... |

6 |
Groups & Graphs, A Macintosh Application for Graph Theory
- Kocay
- 1988
(Show Context)
Citation Context ...em can always be transformed into a 2-cell toroidal or projective planar rotation system. We developed several methods to do this transformation [4] and implemented them in the software Groups&Graphs =-=[8]-=-. In general, a planarity testing algorithm can be modified so that in case of a non-planar graph G it will return a subdivision of K5 or K3,3 in G. We assume that G is a non-planar, 2-connected graph... |

4 |
Embedding a graph into a torus in linear time, Integer programming and combinatorial optimization
- Juvan, Marincek, et al.
- 1995
(Show Context)
Citation Context ... a projective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3-subdivision in the graph G. It significantly simplifies algorithms presented in =-=[7]-=-, [10] and [12]. We then need to consider only the embeddings on the given surface of a K3,3-subdivision, which are much less numerous than those of K5. 1. Introduction We use basic graph-theoretic te... |

3 | Simpler projective plane embedding
- Roth, Myrvold
- 2005
(Show Context)
Citation Context ...planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3-subdivision in the graph G. It significantly simplifies algorithms presented in [7], [10] and =-=[12]-=-. We then need to consider only the embeddings on the given surface of a K3,3-subdivision, which are much less numerous than those of K5. 1. Introduction We use basic graph-theoretic terminology from ... |