Biologists represent evolutionary history of species through phylogenetic trees. Leaves of a phylogenetic tree represent the species and internal vertices represent the extinct ancestors. Given a collection of input phylogenetic trees, a common problem in computational biology is to build a supertree that captures the evolutionary history of all the species in the input trees, and is consistent with each of the input trees. In this document we study the tree compatibility and agreement supertree problems.

Tree compatibility problem is NP-complete but has been shown to be fixed parameter tractable when parametrized by number of input trees. We characterize the compatible supertree problem in terms of triangulation of a structure called the display graph. We also give an alternative characterization in terms of cuts of the display graph. We show how these characterizations are related to characterization given in terms of triangulation of the edge label intersection graph. We then give a characterization of the agreement supertree problem.

In real world data, consistent supertrees do not always exist. Inconsistencies can be dealt with by contraction of edges or removal of taxa. The agreement supertree edge contraction (AST-EC) problem asks if a collection of k rooted trees can be made to agree by contraction of at most p edges. Similarly, the agreement supertree taxon removal (AST-TR) problem asks if a collection of k rooted trees can be made to agree by removal of at most p taxa. We give fixed parameter algorithms for both cases when parametrized by k and p.

We study the long standing conjecture on the perfect phylogeny problem; there exists a function f (r) such that a given collection C of r-state characters is compatible if and only if every f (r) subset of C is compatible. We will show that for r ≥ 2, f (r) ≥ lceil (r/2) rceil * lfloor(r/2)rfloor + 1.