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Deciding whether a collection of unrooted trees is compatible is afundamental problem in phylogenetics. Two different graph-theoreticcharacterizations of tree compatibility have recently been proposed. In one ofthese, tree compatibility is characterized in terms of the existence of aspecific kind of triangulation in a structure known as the display graph. Analternative characterization expresses the tree compatibility problem as achordal graph sandwich problem in a structure known as the edge labelintersection graph. In this paper we show that the characterization using edgelabel intersection graphs transforms to a characterization in terms of minimalcuts of the display graph. We show how these two characterizations are relatedto compatibility of splits. We also show how the characterization in terms ofminimal cuts of display graph is related to the characterization in terms oftriangulation of the display graph.

On Two Graph-Theoretic Characterizations of Tree Compatibility