On compatibility and incompatibility of collections of unrooted phylogenetic trees

Abstract

Let P={T1,…,Tk} be a collection of phylogenetic trees over various subsets of a set of species. For each i∈{1,…,k}, let L(Ti) denote the set of species in tree Ti. A supertree for P is a phylogenetic tree with species set ⋃i=1kL(Ti). The tree compatibility problem asks whether there exists a supertree T for P such that, for each i∈{1,…,k}, Ti can be obtained from T|L(Ti) — the minimal subtree of T spanning L(Ti) — by zero or more contractions of internal edges. If the answer is “yes”, then P is said to be compatible; otherwise, P is incompatible.We characterize compatibility via graph triangulations and tree decompositions. We then study how to make an incompatible collection of trees compatible through edge contraction and tree deletion. Finally, we introduce the notion of a phylogenetic minor to study under which conditions edge contraction, tree removal, and species removal/renaming operations preserve compatibility.