The question whether a partition $mathcal{P}$ and a hierarchy $mathcal{H}$ or a tree-like split system $mathfrak{S}$ are compatible naturally arises in a wide range of classification problems. In the setting of phylogenetic trees, one asks whether the sets of $mathcal{P}$ coincide with leaf sets of connected components obtained by deleting some edges from the tree $T$ that represents $mathcal{H}$ or $mathfrak{S}$, respectively. More generally, we ask whether a refinement $T^*$ of $T$ exists such that $T^*$ and $mathcal{P}$ are compatible. We report several characterizations for (refinements of) hierarchies and split systems that are compatible with (sets of) partitions. In addition, we provide a linear-time algorithm to check whether refinements of trees and a given partition are compatible. The latter problem becomes NP-complete but fixed-parameter tractable if a set of partitions is considered instead of a single partition. We finally explore the close relationship of the concept of compatibility and so-called Fitch maps.
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