For DAGs with a unique LCA for every leaf subset of allowed size, the clusters that survive the simplification operator are exactly those that are inclusion-minimal for some allowed subset, and the deleted vertex set is unique.
Philosophical Transactions of the Royal Society B: Biological Sciences 364(1527):2187–2196, DOI 10
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Characterizing and Transforming DAGs within the I-LCA Framework
For DAGs with a unique LCA for every leaf subset of allowed size, the clusters that survive the simplification operator are exactly those that are inclusion-minimal for some allowed subset, and the deleted vertex set is unique.