Spanning trees in connected graphs with few branch and end vertices

TL;DR

This paper investigates spanning trees in connected graphs with constraints on branch and end vertices, providing new bounds under degree sum conditions relevant to network design, and proves these bounds are optimal.

Contribution

It establishes new bounds on the number of branch and end vertices in spanning trees under degree sum conditions, extending previous results and demonstrating their sharpness.

Findings

Every graph satisfying the degree sum condition has a spanning tree with at most k+1 branch and end vertices.

Such graphs have a spanning tree with at most (k-1)/2 branch vertices.

They also have a spanning tree with a degree sum of branch vertices at most 1.5(k-1).

Abstract

A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree problems arising in the context of optical and centralized terminal networks: finding a spanning tree of G (i) with the minimum number of end vertices, (ii) with the minimum number of branch vertices and (iii) with the minimum degree sum of the branch vertices, motivated by network design problems where junctions are significantly more expensive than simple end- or through-nodes, and are thus to be avoided. We consider: $(\ast)$ connected graphs on $n$ vertices such that $\sigma_2\ge n-k+1$ for some positive integer $k$. In 1976, it was proved (by the author) that every graph satisfying $(\ast)$ has a spanning tree with at most $k$ end vertices. In this paper we first show that every graph satisfying $(\ast)$ has a spanning tree with at most $k+1$ branch and end vertices altogether. The next result states that every graph satisfying $(\ast)$ has a spanning tree with at most $(k-1)/2$ branch vertices. The third result states that every graph satisfying $(\ast)$ has a spanning tree with at most $\frac{3}{2}(k-1)$ degree sum of branch vertices. All results are sharp.