### Abstract

We prove the following theorems: (i) Let G be a graph and let x be a locally 2n-connected vertex. Let {u,v} be a pair of vertices in V(G) - {x} such that uv ∉ E(G), x ∈ N_{G}(u) ∩ N_{G}(v), and N_{G}(x) ⊂ N_{G}(u) ∪ N_{G}(v) ∪ {u, v}. Then if G + uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs described. (ii) Let G be a (2n + l)-connected graph. Let {u,v,x} be a three-vertex subset of V(G) such that uv ∉ E(G), x ∈ N_{G}(u) ∩ N_{G}(v), and N_{G}(x) ⊂ N_{G}(u) ∪ N_{G}(v) ∪ {u,v}. If G + uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs described.

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

Pages (from-to) | 171-175 |

Number of pages | 5 |

Journal | Australasian Journal of Combinatorics |

Volume | 32 |

Publication status | Published - 2005 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*32*, 171-175.

**Note on a closure concept and matching extension.** / Nishimura, Tsuyoshi.

Research output: Contribution to journal › Article

*Australasian Journal of Combinatorics*, vol. 32, pp. 171-175.

}

TY - JOUR

T1 - Note on a closure concept and matching extension

AU - Nishimura, Tsuyoshi

PY - 2005

Y1 - 2005

N2 - We prove the following theorems: (i) Let G be a graph and let x be a locally 2n-connected vertex. Let {u,v} be a pair of vertices in V(G) - {x} such that uv ∉ E(G), x ∈ NG(u) ∩ NG(v), and NG(x) ⊂ NG(u) ∪ NG(v) ∪ {u, v}. Then if G + uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs described. (ii) Let G be a (2n + l)-connected graph. Let {u,v,x} be a three-vertex subset of V(G) such that uv ∉ E(G), x ∈ NG(u) ∩ NG(v), and NG(x) ⊂ NG(u) ∪ NG(v) ∪ {u,v}. If G + uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs described.

AB - We prove the following theorems: (i) Let G be a graph and let x be a locally 2n-connected vertex. Let {u,v} be a pair of vertices in V(G) - {x} such that uv ∉ E(G), x ∈ NG(u) ∩ NG(v), and NG(x) ⊂ NG(u) ∪ NG(v) ∪ {u, v}. Then if G + uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs described. (ii) Let G be a (2n + l)-connected graph. Let {u,v,x} be a three-vertex subset of V(G) such that uv ∉ E(G), x ∈ NG(u) ∩ NG(v), and NG(x) ⊂ NG(u) ∪ NG(v) ∪ {u,v}. If G + uv is n-extendable, then G is n-extendable or G is a member of the exceptional family F of graphs described.

UR - http://www.scopus.com/inward/record.url?scp=84885904840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885904840&partnerID=8YFLogxK

M3 - Article

VL - 32

SP - 171

EP - 175

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

ER -