### 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 |
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Pages (from-to) | 143-159 |

Number of pages | 17 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 31 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

### Keywords

- closure
- k-ended tree
- spanning tree

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discussiones Mathematicae - Graph Theory*,

*31*(1), 143-159. https://doi.org/10.7151/dmgt.1534