Closure for Spanning Trees with k-Ended Stems

Zheng Yan

Department of Computer and Information Sciences, Ibaraki University, Hitachi, Japan.

DOI: 10.4236/ojdm.2014.43008                

Abstract

Let T be a tree. The set of leaves of Τ is denoted by Leaf(Τ). The subtree Τ—Leaf(Τ) of T is called the stem of Τ. A stem is called a k-ended stem if it has at most k-leaves in it. In this paper, we prove the following theorem. Let G be a connected graph and k≥2 be an integer. Let u and ν be a pair of nonadjacent vertices in G. Suppose that |N_G(u)∪N_G(v)|≥|G|-k-1. Then G has a spanning tree with k-ended stem if and only if G+uv has a spanning tree with k-ended stem. Moreover, the condition on |N_G(u)∪N_G(v)| is sharp.

Keywords

Closure, Spanning Tree, Stem, k-End Stem

Conflicts of Interest

The authors declare no conflicts of interest.

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