Quartets in Weakly Compatible Split Systems

Stefan Gruenewald
CAS-MPG Partner Institute for Computational Biology, Shanghai, China

2011/12/9 Fri 4PM-5PM (Room 3433)

A phylogenetic (i.e evolutionary) tree can be interpreted as a compatible split system, that is a collection of bipartitions of a finite set X such that, for all four elements of X, there are no two bipartitions in the collection that induce different splits of those four elements into two pairs. Such a split of a 4-set into two 2-sets is called a quartet, and a split system is said to display a quartet, if there is at least one split in the system that induces this quartet. In phylogenetics, it is often useful to allow more general than compatible split systems, in order to display contradicting signals in the data or to find evidence for reticulate evolution. One natural such generalization are weakly compatible split systems, where for every 4-set at most two of the three possible quartets are allowed to be displayed. The split decomposition algorithm (implemented in the Splitstree software) is a successful tool to construct weakly compatible split systems from distance data. However, weakly compatible split systems are not as well understood as compatible ones. For example, maximal compatible split systems, i.e. compatible split systems which become incompatible whenever a new split is added, correspond to binary trees and display one quartet for every 4-set. In contrast, maximal weakly compatible split systems often display less than the two quartets per 4-set that are allowed by definition. Indeed there are examples where no quartet is displayed for almost all 4-sets. This leaves the question what is the minimum cardinality of maximal weakly compatible split systems for given cardinality of X.

In my talk I will introduce weakly compatible split systems and explain their relevance for phylogenetics, and I will present upper and lower bounds for the smallest number of quartets in maximal weakly compatible split systems.