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June, 2021 Spanning Trees with Few Peripheral Branch Vertices
Pham Hoang Ha, Dang Dinh Hanh, Nguyen Thanh Loan
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Taiwanese J. Math. 25(3): 435-447 (June, 2021). DOI: 10.11650/tjm/201201

Abstract

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. The set of leaves of $T$ is denoted by $L(T)$ and the set of branch vertices of $T$ is denoted by $B(T)$. For two distinct vertices $u$, $v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v$. Let $T$ be a tree with $B(T) \neq \emptyset$, for each vertex $x \in L(T)$, set $y_x \in B(T)$ such that $(V(P_T[x,y_x]) \setminus \{y_x\}) \cap B(T) = \emptyset$. We delete $V(P_T[x,y_x]) \setminus \{y_x\}$ from $T$ for all $x \in L(T)$. The resulting graph is a subtree of $T$ and is denoted by $\operatorname{R\_Stem}(T)$. It is called the reducible stem of $T$. A leaf of $\operatorname{R\_Stem}(T)$ is called a peripheral branch vertex of $T$. In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph $G$ to have a spanning tree with few peripheral branch vertices.

Funding Statement

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2018.03.

Acknowledgments

A part of work was completed during a stay of the first named author at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank this institution for financial support and hospitality.

Citation

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Pham Hoang Ha. Dang Dinh Hanh. Nguyen Thanh Loan. "Spanning Trees with Few Peripheral Branch Vertices." Taiwanese J. Math. 25 (3) 435 - 447, June, 2021. https://doi.org/10.11650/tjm/201201

Information

Received: 14 August 2020; Revised: 20 November 2020; Accepted: 29 November 2020; Published: June, 2021
First available in Project Euclid: 9 December 2020

Digital Object Identifier: 10.11650/tjm/201201

Subjects:
Primary: 05C05, 05C07, 05C69

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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