Perfect taxon sampling and phylogenetically decisive taxon coverage

In a recent study, Steel and Sanderson defined and characterized phylogenetically decisive sets of taxon sets. A set is called phylogenetically decisive if regardless of the trees chosen for each of its taxon sets, as long as these trees are compatible with one another, their supertree is always unique. This implies that the sampled taxon sets always lead to a unique supertree, regardless what tree they support (as long as the trees of all the taxon sets are compatible) -- which is why this setting can be referred to as perfect taxon sampling. However, the complexity of the decision problem to determine whether a set of taxon sets is phylogenetically decisive remained unknown. This problem was one of the Penny Ante prize questions of the Annual New Zealand Phylogenetics Meeting in 2012. In this paper, we explain phylogenetic decisiveness and demonstrate a new characterization, which then leads to a polynomial time algorithm for the case where the number of taxon sets under consideration is polynomial in the number of taxa - both for the (simpler) rooted tree case as well as for the (more complicated) unrooted tree case.