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