Compatibility of Partitions, Hierarchies, and Split Systems
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The question whether a partition 𝒫 and a hierarchy ℋ or a tree-like split system 𝔖 are compatible naturally arises in a wide range of classification problems. In the setting of phylogenetic trees, one asks whether the sets of 𝒫 coincide with leaf sets of connected components obtained by deleting some edges from the tree T that represents ℋ or 𝔖, respectively. More generally, we ask whether a refinement T^* of T exists such that T^* and 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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