### Abstract

A cubic graph G is cyclically 5-connected if G is simple, 3-connected, has at least 10 vertices and for every set F of edges of size at most four, at most one component of G\F contains circuits. We prove that if G and H are cyclically 5-connected cubic graphs and H topologically contains G, then either G and H are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph G^{′} such that H topologically contains G^{′} and G^{′} is obtained from G in one of the following two ways. Either G^{′} is obtained from G by subdividing two distinct edges of G and joining the two new vertices by an edge, or G^{′} is obtained from G by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from G in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of H we are able to eliminate the second construction. We also prove versions of both of these results when G is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case G^{′} is required to be almost cyclically 5-connected and to have fewer circuits of length four than G. In particular, if G has at most one circuit of length four, then G^{′} is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs G^{′} are more complicated.

Original language | English (US) |
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Pages (from-to) | 132-167 |

Number of pages | 36 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 125 |

DOIs | |

State | Published - Jul 2017 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Cyclically 5-connected cubic graph
- Generation theorem

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*125*, 132-167. https://doi.org/10.1016/j.jctb.2017.03.003