### Abstract

Let G = (V, E) be a graph and r: V → Z_{+}. An r-detachment of G is a graph H obtained by 'splitting' each vertex v ε V into r(v) vertices, called the pieces of v in H. Every edge uv ε E corresponds to an edge of H connecting some piece of u to some piece of v. An r-degree specification for G is a function f on V, such that, for each vertex v ε V, f (v) is a partition of d(v) into r(v) positive integers. An f-detachment of G is an r-detachment H in which the degrees in H of the pieces of each v ε V are given by f (v). Crispin Nash-Williams [3] obtained necessary and sufficient conditions for a graph to have a k-edge-connected r-detachment or f-detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs.

Original language | English |
---|---|

Pages (from-to) | 17-37 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 87 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2003 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*87*(1), 17-37. https://doi.org/10.1016/S0095-8956(02)00026-6

**Non-separable detachments of graphs.** / Jackson, Bill; Jordán, T.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 87, no. 1, pp. 17-37. https://doi.org/10.1016/S0095-8956(02)00026-6

}

TY - JOUR

T1 - Non-separable detachments of graphs

AU - Jackson, Bill

AU - Jordán, T.

PY - 2003/1

Y1 - 2003/1

N2 - Let G = (V, E) be a graph and r: V → Z+. An r-detachment of G is a graph H obtained by 'splitting' each vertex v ε V into r(v) vertices, called the pieces of v in H. Every edge uv ε E corresponds to an edge of H connecting some piece of u to some piece of v. An r-degree specification for G is a function f on V, such that, for each vertex v ε V, f (v) is a partition of d(v) into r(v) positive integers. An f-detachment of G is an r-detachment H in which the degrees in H of the pieces of each v ε V are given by f (v). Crispin Nash-Williams [3] obtained necessary and sufficient conditions for a graph to have a k-edge-connected r-detachment or f-detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs.

AB - Let G = (V, E) be a graph and r: V → Z+. An r-detachment of G is a graph H obtained by 'splitting' each vertex v ε V into r(v) vertices, called the pieces of v in H. Every edge uv ε E corresponds to an edge of H connecting some piece of u to some piece of v. An r-degree specification for G is a function f on V, such that, for each vertex v ε V, f (v) is a partition of d(v) into r(v) positive integers. An f-detachment of G is an r-detachment H in which the degrees in H of the pieces of each v ε V are given by f (v). Crispin Nash-Williams [3] obtained necessary and sufficient conditions for a graph to have a k-edge-connected r-detachment or f-detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs.

UR - http://www.scopus.com/inward/record.url?scp=0037288729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037288729&partnerID=8YFLogxK

U2 - 10.1016/S0095-8956(02)00026-6

DO - 10.1016/S0095-8956(02)00026-6

M3 - Article

AN - SCOPUS:0037288729

VL - 87

SP - 17

EP - 37

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -