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Phylogenetically decisive sets of taxon sets have the property that if treesare chosen for each of their elements, as long as these trees are compatible,the resulting supertree is unique. This means that as long as the treesdescribing the phylogenetic relationships of the (input) species sets arecompatible, they can only be combined into a common supertree in precisely oneway. This setting is sometimes also referred to as \enquote{perfect taxonsampling}. While for rooted trees, the decision if a given set of input taxonsets is phylogenetically decisive can be made in polynomial time, the decisionproblem to determine whether a set of taxon sets is phylogenetically decisiveconcerning \emph{unrooted} trees is unfortunately coNP-complete. This showsthat recognizing such sets is often difficult. In this paper, we explainphylogenetic decisiveness and introduce a class of input taxon sets, namelyso-called \emph{fixing taxon traceable} sets, which are guaranteed to bephylogenetically decisive and which can be recognized in polynomial time. Usingboth combinatorial approaches as well as simulations, we compare properties offixing taxon traceability and phylogenetic decisiveness, e.g., by derivinglower and upper bounds for the number of quadruple sets needed in the input setfor each of these properties. In particular, we correct an erroneous lowerbound concerning phylogenetic decisiveness from the literature.  We have implemented the algorithm to determine if a given set of taxon setsis fixing taxon traceable in \textsf{R} and made our software package\verb+FixingTaxonTraceR+ publicly available.

Perfect taxon sampling and fixing taxon traceability: Introducing a class of phylogenetically decisive taxon sets