On low treewidth graphs and supertrees

TL;DR

This paper investigates the relationship between the treewidth of display graphs and the compatibility of unrooted phylogenetic trees, providing new polynomial-time algorithms for cases with low treewidth.

Contribution

It proves that low treewidth (at most 2) of display graphs guarantees compatibility and offers a polynomial-time construction of supertrees in this case.

Findings

Display graphs with treewidth ≤ 2 are always compatible.

Polynomial-time algorithm for constructing supertrees when treewidth ≤ 2.

Treewidth 3 display graphs can be compatible or incompatible, so treewidth alone is not decisive.

Abstract

Compatibility of unrooted phylogenetic trees is a well studied problem in phylogenetics. It asks to determine whether for a set of k input trees there exists a larger tree (called a supertree) that contains the topologies of all k input trees. When any such supertree exists we call the instance compatible and otherwise incompatible. It is known that the problem is NP-hard and FPT, although a constructive FPT algorithm is not known. It has been shown that whenever the treewidth of an auxiliary structure known as the display graph is strictly larger than the number of input trees, the instance is incompatible. Here we show that whenever the treewidth of the display graph is at most 2, the instance is compatible. Furthermore, we give a polynomial-time algorithm to construct a supertree in this case. Finally, we demonstrate both compatible and incompatible instances that have display graphs with treewidth 3, highlighting that the treewidth of the display graph is (on its own) not sufficient to determine compatibility.