Contraction of graphs and spanning k-end trees

TL;DR

This paper investigates how graph contractions affect the existence of spanning k-end trees, proving that certain contractions preserve the property under specific conditions.

Contribution

It introduces new theorems linking graph contractions to the preservation of spanning k-end trees in connected graphs.

Findings

Contraction along specific edges preserves spanning k-end trees.

Connected graphs with spanning k-end trees retain this property after contraction under certain size conditions.

Provides theoretical foundations for graph contraction operations related to spanning trees.

Abstract

A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end vertices and deleting all edges between every two vertex except one edge to gain again a simple graph and is denoted bye G/e. In this paper we prove some theorems related to a graph and its contraction. For example we prove the following theorem. If G is a connected graph that has a spanning k-end tree and |V (G)| > K + 1 then there exist an edge e such G/e has a spanning k-end tree.