An Algorithm for the Decomposition of Complete Graph into Minimum Number of Edge-disjoint Trees

TL;DR

This paper presents an algorithm for decomposing a complete graph into the minimum number of edge-disjoint trees, achieving optimal decomposition with efficient linear time complexity.

Contribution

It introduces a method to decompose complete graphs into the minimum number of edge-disjoint trees with proven optimality and linear time complexity.

Findings

Minimum number of trees needed is loorrac{n}{2}loor.

Decomposition covers all edges with minimal trees.

Algorithm runs in linear time proportional to the number of edges.

Abstract

In this work, we study methodical decomposition of an undirected, unweighted complete graph ($K_n$ of order $n$, size $m$) into minimum number of edge-disjoint trees. We find that $x$, a positive integer, is minimum and $x=\lceil\frac{n}{2}\rceil$ as the edge set of $K_n$ is decomposed into edge-disjoint trees of size sequence $M = \{m_1,m_2,...,m_x\}$ where $m_i\le(n-1)$ and $\Sigma_{i=1}^{x} m_i$ = $\frac{n(n-1)}{2}$. For decomposing the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed algorithm takes total $O(m)$ time.