Improved lower bounds on the compatibility of quartets, triplets, and multi-state characters

• Authors: Brad Shutters;Sudheer Vakati;David Fernández-Baca
• Venue: WABI'12 Proceedings of the 12th international conference on Algorithms in Bioinformatics
• Year: 2012

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r≥2, there exists an incompatible set C of $\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ r-state characters such that every proper subset of C is compatible. Thus, f(r) ≥ $\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1$ for every r≥2. This improves the previous lower bound of f(r)≥r given by Meacham (1983), and generalizes the construction showing that f(4)≥5 given by Habib and To (2011). We prove our result via a result on quartet compatibility that may be of independent interest: For every integer n≥4, there exists an incompatible set Q of $\lfloor\frac{n-2}{2}\rfloor\cdot\lceil\frac{n-2}{2}\rceil + 1$ quartets over n labels such that every proper subset of Q is compatible. We contrast this with a result on the compatibility of triplets: For every n≥3, if R is an incompatible set of more than n−1 triplets over n labels, then some proper subset of R is incompatible. We show this bound is tight by exhibiting, for every n≥3, a set of n−1 triplets over n taxa such that R is incompatible, but every proper subset of R is compatible.