No abstract

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 Ω(r2)r-state characters such that every proper subset of C is compatible. This improves the previous lower bound of f(r)≥r given by Meacham (1983), and f(4)≥5 given by Habib and To (2011). For the case when r=3, Lam, Gusfield and Sridhar (2011) recently showed that f(3)=3. We give an independent proof of this result and completely characterize the sets of pairwise compatible 3-state characters by a single forbidden intersection pattern.Our lower bound on f(r) is proven via a result on quartet compatibility that may be of independent interest: For every n≥4, there exists an incompatible set Q of Ω(n2) quartets over n labels such that every proper subset of Q is compatible. We show that such a set of quartets can have size at most 3 when n=5, and at most O(n3) for arbitrary n. We contrast our results on quartets with the case of rooted 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.

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the "strict" sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (≈ 1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree.

No abstract

Lam, Gusfield, and Sridhar (2009) showed that a set of three-state characters has a perfect phylogeny if and only if every subset of three characters has a perfect phylogeny. They also gave a complete characterization of the sets of three three-state characters that do not have a perfect phylogeny. However, it is not clear from their characterization how to find a subset of three characters that does not have a perfect phylogeny without testing all triples of characters. In this note, we build upon their result by giving a simple characterization of when a set of three-state characters does not have a perfect phylogeny that can be inferred from testing all pairs of characters.