# Selecting optimal minimum spanning trees that share a topological correspondence with phylogenetic trees

## TL;DR

This paper addresses the indeterminacy in MST-based phylogenetic tree construction methods by introducing vertex-ranked MSTs and provides an efficient algorithm for finding the minimum-leaf vertex-ranked MST in certain graphs.

## Contribution

It introduces vertex-ranked MSTs to resolve MST indeterminacy and presents a polynomial-time algorithm for the MLVRMST problem in tree-additive weighted graphs.

## Key findings

- Resolved MST indeterminacy with vertex-ranked MSTs.
- Developed a polynomial-time algorithm for MLVRMST.
- Proved correctness for graphs with tree-additive distances.

## Abstract

Choi et. al (2011) introduced a minimum spanning tree (MST)-based method called CLGrouping, for constructing tree-structured probabilistic graphical models, a statistical framework that is commonly used for inferring phylogenetic trees. While CLGrouping works correctly if there is a unique MST, we observe an indeterminacy in the method in the case that there are multiple MSTs. In this work we remove this indeterminacy by introducing so-called vertex-ranked MSTs. We note that the effectiveness of CLGrouping is inversely related to the number of leaves in the MST. This motivates the problem of finding a vertex-ranked MST with the minimum number of leaves (MLVRMST). We provide a polynomial time algorithm for the MLVRMST problem, and prove its correctness for graphs whose edges are weighted with tree-additive distances.