Note on a closure concept and matching extension

Tsuyoshi Nishimura

Note on a closure concept and matching extension

Research output: Contribution to journal › Article › peer-review

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 Dec 1

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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title = "Note on a closure concept and matching extension",

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 ∈ 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.",

author = "Tsuyoshi Nishimura",

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AU - Nishimura, Tsuyoshi

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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.

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