A Stronger Lower Bound on Parametric Minimum Spanning Trees

TL;DR

This paper establishes a new lower bound on the number of distinct minimum spanning trees in a parameterized graph, showing it can be as large as Omega(m log n) as the parameter varies.

Contribution

It provides a stronger theoretical lower bound on the number of parametric minimum spanning trees, advancing understanding of their combinatorial complexity.

Findings

Number of distinct MSTs can be Omega(m log n)

Lower bound applies to graphs with linear parameter functions

Results improve previous bounds on parametric MST complexity

Abstract

We prove that, for an undirected graph with $n$ vertices and $m$ edges, each labeled with a linear function of a parameter $\lambda$, the number of different minimum spanning trees obtained as the parameter varies can be $\Omega(m\log n)$.