### Abstract

We introduce a topological graph parameter s(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with s(G)=1,2,3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then s(H)=s(G), that s(K n )=n-1, and that if H is the suspension of G, then s(H)=s(G)+1. Furthermore, we show that µ(G)=s(G) + 2 for each graph G. Here µ(G) is the graph parameter introduced by Colin de Verdière in [2].

Original language | English |
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Pages (from-to) | 337-361 |

Journal | Combinatorica |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 |

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## Cite this

Holst, van der, H., & Pendavingh, R. A. (2009). On a graph property generalizing planarity and flatness.

*Combinatorica*,*29*(3), 337-361. https://doi.org/10.1007/s00493-009-2219-6