Closure for spanning trees and distant area

Jun Fujisawa, Akira Saito, Ingo Schiermeyer

Research output: Contribution to journalArticle

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 + 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.

Original languageEnglish
Pages (from-to)143-159
Number of pages17
JournalDiscussiones Mathematicae - Graph Theory
Volume31
Issue number1
DOIs
Publication statusPublished - 2011

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Keywords

  • closure
  • k-ended tree
  • spanning tree

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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