# Closure for spanning trees and distant area

• Volume: 31, Issue: 1, page 143-159
• ISSN: 2083-5892

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## Abstract

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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 $de{g}_{G}u+de{g}_{G}v\ge n-1$, 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 $de{g}_{G}u+de{g}_{G}v$ and the structure of the distant area for u and v. We prove that if the distant area contains ${K}_{r}$, we can relax the lower bound of $de{g}_{G}u+de{g}_{G}v$ 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.

## How to cite

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Jun Fujisawa, Akira Saito, and Ingo Schiermeyer. "Closure for spanning trees and distant area." Discussiones Mathematicae Graph Theory 31.1 (2011): 143-159. <http://eudml.org/doc/271024>.

@article{JunFujisawa2011,
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 $deg_G u + deg_G v ≥ n-1$, 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 $K_r$, 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.},
author = {Jun Fujisawa, Akira Saito, Ingo Schiermeyer},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {spanning tree; k-ended tree; closure; -ended tree, closure},
language = {eng},
number = {1},
pages = {143-159},
title = {Closure for spanning trees and distant area},
url = {http://eudml.org/doc/271024},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Jun Fujisawa
AU - Akira Saito
AU - Ingo Schiermeyer
TI - Closure for spanning trees and distant area
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 1
SP - 143
EP - 159
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 $deg_G u + deg_G v ≥ n-1$, 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 $K_r$, 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.
LA - eng
KW - spanning tree; k-ended tree; closure; -ended tree, closure
UR - http://eudml.org/doc/271024
ER -

## References

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1. [1] J.A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15 (1976) 111-135, doi: 10.1016/0012-365X(76)90078-9. Zbl0331.05138
2. [2] H.J. Broersma and I. Schiermeyer, A closure concept based on neighborhood unions of independent triples, Discrete Math. 124 (1994) 37-47, doi: 10.1016/0012-365X(92)00049-W. Zbl0789.05059
3. [3] H. Broersma and H. Tuinstra, Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227-237, doi: 10.1002/(SICI)1097-0118(199812)29:4<227::AID-JGT2>3.0.CO;2-W Zbl0919.05017
4. [4] G. Chartrand and L. Lesniak, Graphs & Digraphs (4th ed.), (Chapman and Hall/CRC, Boca Raton, Florida, U.S.A. 2005).
5. [5] Y.J. Zhu, F. Tian and X.T. Deng, Further consideration on the Bondy-Chvátal closure theorems, in: Graph Theory, Combinatorics, Algorithms, and Applications (San Francisco, CA, 1989), 518-524. Zbl0746.05041

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