Strictly chordal graphs are leaf powers

A fundamental problem in computational biology is the phylogeny reconstruction for a set of specific organisms. One of the graph theoretical approaches is to construct a similarity graph on the set of organisms where adjacency indicates evolutionary closeness, and then to reconstruct a phylogeny by computing a tree interconnecting the organisms such that leaves in the tree are labeled by the organisms and every organism appears as a leaf in the tree. The similarity graph is simple and undirected. For any pair of adjacent organisms in the similarity graph, their distance in the output tree, which is measured by the number of edges on the path connecting them, must be less than some pre-specified bound. This is known as the problem of recognizing leaf powers and computing leaf roots. Graphs that are leaf powers are known to be chordal. It is shown in this paper that all strictly chordal graphs are leaf powers and a linear time algorithm is presented to compute a leaf root for any given strictly chordal graph. An intermediate root-and-power problem, the Steiner root problem, is also examined.