### Abstract

A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with degG u + deg_{G} v ≥ n - 1, G has a spanning k-ended tree if and only if G + uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on deg_{G} u + deg_{G} v and the structure of the distant area for u and v. We prove that if the distant area contains Kr, we can relax the lower bound of deg_{G} u+deg_{G} v from n - 1 to n - r. And if the distant area itself is a complete graph and G is 2-connected, we can entirely remove the degree sum condition.

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

Pages (from-to) | 143-159 |

Number of pages | 17 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 31 |

Issue number | 1 |

Publication status | Published - 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- closure
- k-ended tree
- spanning tree

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discussiones Mathematicae - Graph Theory*,

*31*(1), 143-159.

**Closure for spanning trees and distant area.** / Fujisawa, Jun; Saito, Akira; Schiermeyer, Ingo.

Research output: Contribution to journal › Article

*Discussiones Mathematicae - Graph Theory*, vol. 31, no. 1, pp. 143-159.

}

TY - JOUR

T1 - Closure for spanning trees and distant area

AU - Fujisawa, Jun

AU - Saito, Akira

AU - Schiermeyer, Ingo

PY - 2011

Y1 - 2011

N2 - A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with degG u + degG v ≥ n - 1, G has a spanning k-ended tree if and only if G + uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on degG u + degG v and the structure of the distant area for u and v. We prove that if the distant area contains Kr, we can relax the lower bound of degG u+degG v from n - 1 to n - r. And if the distant area itself is a complete graph and G is 2-connected, we can entirely remove the degree sum condition.

AB - A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with degG u + degG v ≥ n - 1, G has a spanning k-ended tree if and only if G + uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on degG u + degG v and the structure of the distant area for u and v. We prove that if the distant area contains Kr, we can relax the lower bound of degG u+degG v from n - 1 to n - r. And if the distant area itself is a complete graph and G is 2-connected, we can entirely remove the degree sum condition.

KW - closure

KW - k-ended tree

KW - spanning tree

UR - http://www.scopus.com/inward/record.url?scp=78751549659&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751549659&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:78751549659

VL - 31

SP - 143

EP - 159

JO - Discussiones Mathematicae - Graph Theory

JF - Discussiones Mathematicae - Graph Theory

SN - 1234-3099

IS - 1

ER -