Rooted directed path graphs are leaf powers

Leaf powers are a graph class which has been introduced to model the problem of reconstructing phylogenetic trees. A graph G=(V,E)G=(V,E) is called kk-leaf power if it admits a kk-leaf root, i.e., a tree TT with leaves VV such that uvuv is an edge in GG if and only if the distance between uu and vv in TT is at most kk. Moroever, a graph is simply called leaf power if it is a kk-leaf power for some k∈Nk∈N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a given graph is a kk-leaf power.We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a well-known graph class with absolutely different motivations. After this, we provide the largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path graphs.Subsequently, we study the leaf rank problem, the algorithmic challenge of determining the minimum kk for which a given graph is a kk-leaf power. Firstly, we give a lower bound on the leaf rank of a graph in terms of the complexity of its separators. Secondly, we use this measure to show that the leaf rank is unbounded on both the class of ptolemaic and the class of unit interval graphs. Finally, we provide efficient algorithms to compute 2|V|2|V|-leaf roots for given ptolemaic or (unit) interval graphs G=(V,E)G=(V,E).