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Leaf powers are a graph class which has been introduced to model the problem of reconstructing phylogenetic trees. A graph G=(V,E) is called k-leaf power if it admits a k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a k-leaf power for some [email protected]?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 k-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 k for which a given graph is a k-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|-leaf roots for given ptolemaic or (unit) interval graphs G=(V,E).

In a graph, a vertex is simplicial if its neighborhood is a clique. For an integer k>=1, a graph G=(V"G,E"G) is the k-simplicial power of a graph H=(V"H,E"H) (H a root graph of G) if V"G is the set of all simplicial vertices of H, and for all distinct vertices x and y in V"G, xy@?E"G if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k@?5, k-leaf powers can be recognized in linear time, and for k@?4, structural characterizations are known. For k>=6, the recognition and characterization problems of k-leaf powers are still open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs.

Ptolemaic graphs are those satisfying the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. It can also be seen as a natural generalization of block graphs (and hence trees). In this paper, we first state a laminar structure of cliques, which leads to its canonical tree representation. This result is a translation of γ-acyclicity which appears in the context of relational database schemes. The tree representation gives a simple intersection model of ptolemaic graphs, and it is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence the recognition and the graph isomorphism for ptolemaic graphs can be solved in linear time. Using the tree representation, we also give an efficient algorithm for the Hamiltonian cycle problem.