### Abstract

We prove that if H and H’ are subgraphs of a graph G, and both are isomorphic to subdivisions of K_{5} or K_{3, 3}, then the following are equivalent: (i) there is a sequence H = H_{1}, H_{2}, ⋯, H_{k} = H’ of subgraphs of G, each isomorphic to a subdivision of K_{5} or K_{3, 3} and each differing only a “small amounty” from its predecessor; (ii) H and H’ are not “separated” in G by a vertex separation of order ≤ 3. This is a lemma for use in a future paper concerning linkless embeddings of graphs in 3-space.

Original language | English (US) |
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Pages (from-to) | 127-154 |

Number of pages | 28 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1995 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Robertson, N., Seymour, P. D., & Thomas, R. (1995). Kuratowski Chains.

*Journal of Combinatorial Theory, Series B*,*64*(2), 127-154. https://doi.org/10.1006/jctb.1995.1030