Bounds on the Treewidth of Level-k Rooted Phylogenetic Networks

Abstract

Phylogenetic networks are directed acyclic graphs that depict the genomic evolution of related taxa. Reticulation nodes in such networks (nodes with more than one parent) represent reticulate evolutionary events, such as recombination, reassortment, hybridization, or horizontal gene transfer. Typically, the complexity of a phylogenetic network is expressed in terms of its level, i.e., the maximum number of edges that are required to be removed from each biconnected component of the phylogenetic network to turn it into a tree. Here, we study the relationship between the level of a phylogenetic network and another popular graph complexity parameter - treewidth. We show a $\frac{k+3}{2}$ upper bound on the treewidth of level-$k$ phylogenetic networks and an improved $(1/3 + δ) k$ upper bound for large $k$. These bounds imply that many computational problems on phylogenetic networks, such as the small parsimony problem or some variants of phylogenetic diversity maximization, are polynomial-time solvable on level-$k$ networks with constant $k$. Our first bound is applicable to any $k$, and it allows us to construct an explicit tree decomposition of width $\frac{k+3}{2}$ that can be used to analyze phylogenetic networks generated by tools like SNAQ that guarantee bounded network level. Finally, we show a $k/13$ lower bound on the maximum treewidth among level-$k$ phylogenetic networks for large enough $k$ based on expander graphs.