# Tree decomposition of Reeb graphs, parametrized complexity, and applications to phylogenetics

## TL;DR

This paper demonstrates that Reeb graphs can be decomposed into tree structures, enabling classification and efficient isomorphism testing, with applications to phylogenetics and network analysis.

## Contribution

It introduces a tree decomposition framework for Reeb graphs, showing their classification up to isomorphism and fixed parameter tractability for the isomorphism problem.

## Key findings

- Reeb graphs with n leaves and Betti number s decompose into at most 2^s trees.
- The isomorphism problem for Reeb graphs is fixed parameter tractable based on Betti number.
- Ordered Reeb graphs serve as models for time consistent phylogenetic networks.

## Abstract

Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of \textit{ordered Reeb graphs}, every Reeb graph with $n$ leaves and first Betti number $s$, is equal to a coproduct of at most $2^s$ trees with $(n + s)$ leaves. Reeb graphs are therefore classified up to isomorphism by their tree decomposition. An implication of this result, is that the isomorphism problem for Reeb graphs is fixed parameter tractable when the parameter is the first Betti number. We propose ordered Reeb graphs as a model for time consistent phylogenetic networks and propose a certain Hausdorff distance as a metric on these structures.