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.
Trends in Genetics 16(5):227–231, DOI 10.1016/S0168-9525(00)02005-9
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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.