Unique Perfect Phylogeny Is NP-Hard

We answer, in the affirmative, the following question proposed by Mike Steel as a $100 challenge: "Is the following problem NP-hard? Given a ternary phylogenetic X-tree T and a collection Q of quartet subtrees on X, is T the only tree that displays Q?" [28, 29] As a particular consequence of this, we show that the unique chordal sandwich problem is also NP-hard.

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