Determining the Consistency of Resolved Triplets and Fan Triplets.

The [Formula: see text] Consistency problem takes as input two sets [Formula: see text] and [Formula: see text] of resolved triplets and two sets [Formula: see text] and [Formula: see text] of fan triplets, and asks for a distinctly leaf-labeled tree that contains all elements in [Formula: see text] and no elements in [Formula: see text] as embedded subtrees, if such a tree exists. This article presents a detailed characterization of how the computational complexity of the problem changes under various restrictions. Our main result is an efficient algorithm for dense inputs satisfying [Formula: see text] whose running time is linear in the size of the input and therefore optimal.