A note on independence number, connectivity and $k$-ended tree

TL;DR

This paper presents a simple proof of a theorem linking independence number, connectivity, and the existence of a k-ended tree covering a specific vertex subset in a connected graph, with the condition proven to be sharp.

Contribution

The paper provides a straightforward proof of a theorem connecting independence number, connectivity, and k-ended trees, establishing the sharpness of the condition.

Findings

Proved a theorem relating independence number and connectivity to k-ended trees.

Established the sharpness of the condition for the existence of such trees.

Simplified the proof of a known result in graph theory.

Abstract

A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\geq 2$). Let $S$ be a vertex subset of $G$ such that $\alpha_{G}(S) \leq k + \kappa_{G}(S)- 1.$ Then, $G$ has a $k$-ended tree which covers $S.$ Moreover, the condition is sharp.