2 4 Se p 20 09 Note on the 4 - and 5 - leaf powers ∗

Motivated by the problem of reconstructing evolutionary history, Nishimura et al. defined k-leaf powers as the class of graphs G = (V,E) which has a k-leaf root T , i.e., T is a tree such that the vertices of G are exactly the leaves of T and two vertices in V are adjacent in G if and only if their distance in T is at most k. It is known that leaf powers are chordal graphs. Brandstädt and Le proved that every k-leaf power is a (k + 2)-leaf power and every 3-leaf power is a k-leaf power for k ≥ 3. They asked whether a k-leaf power is also a (k + 1)-leaf power for any k ≥ 4. Fellows et al. gave an example of a 4-leaf power which is not a 5-leaf power. It is interesting to find all the graphs which have both 4-leaf roots and 5-leaf roots. In this paper, we prove that, if G is a 4-leaf power with L(G) 6= ∅, then G is also a 5-leaf power, where L(G) denotes the set of leaves of G.