Abstract
Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in linear time. Next we show that if a phylogenetic network displays (i.e. topologically embeds) a phylogenetic tree, the treewidth of their display graph is bounded by a function of the treewidth of the original network (and also by various other parameters). In fact, using a bramble argument we show that this treewidth bound is sharp up to an additive term of 1. We leverage this bound to give an FPT algorithm, parameterized by treewidth, for determining whether a network displays a tree, which is an intensively-studied problem in the field. We conclude with a discussion on the future use of display graphs and treewidth in phylogenetics.
Highlights
A phylogenetic tree on a set of species X is a tree whose leaves are bijectively labelled by X
We focus on unrooted, binary trees: internal nodes all have degree 3, and there is no direction on the edges of the tree
: given a cubic graph G, do there exist two unrooted binary phylogenetic trees T1, T2 on the same set of taxa X such that G is the display graph D(T1, T2) of T1 and T2? We prove that the problem is NP-hard, by providing an equivalence with the NP-hard TreeArboricity problem [13]
Summary
A phylogenetic tree on a set of species (or, more abstractly, taxa) X is a tree whose leaves are bijectively labelled by X. Longer part of the article, we turn our attention to display graphs formed by merging an unrooted binary phylogenetic tree T with an unrooted binary phylogenetic network N , both on the same set of taxa X The latter is an undirected graph where internal nodes have degree 3 and leaves, as usual, are bijectively labelled by X. Following [34] we use these upper bounds to give a compact MSOL-based fixed-parameter tractable algorithm for the NP-hard problem of determining whether an unrooted network N displays T , under various parameterizations This problem, in the rooted setting, continues to attract significant interest in the phylogenetics literature (see [26, 44, 45] for relevant references). In the final part of the article we reflect on the potential future use of display graphs and treewidth in phylogenetics, and list a number of open problems
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