On the Comparison of Incompatibility of Split Systems Across Different Numbers of Taxa.

We consider the problem of the minimum number of phylogenetic trees it would take to display all splits in a given set, a problem related to k-compatibility. A set of trees that displays every single possible split is termed a universal tree set. In this note, we find the universal incompatibility U(n), the minimal size of a universal tree set for n taxa. By normalising incompatibility using U(n), one can then compare incompatibility of split systems across different numbers of taxa. We demonstrate this application by comparing two SplitsTree networks derived from archaeal genomes, with different numbers of taxa.