On Two Graph-Theoretic Characterizations of Tree Compatibility

TL;DR

This paper explores two graph-theoretic characterizations of tree compatibility in phylogenetics, linking triangulations, minimal cuts, and split compatibility to better understand unrooted tree compatibility.

Contribution

It establishes the relationship between two recent characterizations of tree compatibility, connecting triangulations, minimal cuts, and split compatibility in display and intersection graphs.

Findings

Characterization via minimal cuts relates to split compatibility.

Triangulation of display graph is connected to minimal cut characterization.

The two characterizations are shown to be fundamentally related.

Abstract

Deciding whether a collection of unrooted trees is compatible is a fundamental problem in phylogenetics. Two different graph-theoretic characterizations of tree compatibility have recently been proposed. In one of these, tree compatibility is characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. An alternative characterization expresses the tree compatibility problem as a chordal graph sandwich problem in a structure known as the edge label intersection graph. In this paper we show that the characterization using edge label intersection graphs transforms to a characterization in terms of minimal cuts of the display graph. We show how these two characterizations are related to compatibility of splits. We also show how the characterization in terms of minimal cuts of display graph is related to the characterization in terms of triangulation of the display graph.