Treewidth distance on phylogenetic trees

In this article we study the treewidth of the \emph{display graph}, an auxiliary graph structure obtained from the fusion of phylogenetic (i.e., evolutionary) trees at their leaves. Earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar. Here we further expand upon this relationship. We analyse a number of reduction rules which are commonly used in the phylogenetics literature to obtain fixed parameter tractable algorithms. In some cases (the \emph{subtree} reduction) the reduction rules behave similarly with respect to treewidth, while others (the \emph{cluster} reduction) behave very differently, and the behaviour of the \emph{chain reduction} is particularly intriguing because of its link with graph separators and forbidden minors. We also show that the gap between treewidth and Tree Bisection and Reconnect (TBR) distance can be infinitely large, and that unlike, for example, planar graphs the treewidth of the display graph can be as much as linear in its number of vertices. On a slightly different note we show that if a display graph is formed from the fusion of a phylogenetic network and a tree, rather than from two trees, the treewidth of the display graph is bounded whenever the tree can be topologically embedded ("displayed") within the network. This opens the door to the formulation of the display problem in Monadic Second Order Logic (MSOL). A number of other auxiliary results are given. We conclude with a discussion and list a number of open problems.