On Leaf Powers

For an integer k, a tree T is a k-leaf root of a finite simple undirected graph G = (V,E) if the set of leaves of T is the vertex set V of G and for any two vertices x, y ∈ V , x 6= y, xy ∈ E if and only if the distance of x and y in T is at most k. Then graph G is a k-leaf power if it has a k-leaf root. G is a leaf power if it is a k-leaf power for some k. This notion was introduced and studied by Nishimura, Ragde and Thilikos; it has its background and motivation in computational biology and phylogeny. In this survey, we describe recent results on leaf powers, variants and generalizations. We discuss the relationship between leaf powers and strongly chordal graphs as well as fixed tolerance NeST graphs, describe some subclasses of leaf powers, give the complete inclusion structure of k-leaf power classes, and describe various characterizations of 3and 4-leaf powers, as well as of distance-hereditary 5-leaf powers. Finally we discuss two variants of the notion of k-leaf power such as (k, l)-leaf powers and exact leaf powers, and we generalize leaf powers (of trees) to simplicial powers of graphs. Most of the presented results are part of joint work, mostly with Van Bang Le and Peter Wagner, but also with Christian Hundt, Federico Mancini, R. Sritharan, and Dieter Rautenbach.