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

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.