Some remarks about leaf roots

Nishimura et al. [On graph powers for leaf-labeled trees, J. Algorithms 42 (2002) 69–108] define a k-leaf root of a graph image as a tree image such that the vertices of G are exactly the leaves of T and two vertices in image are adjacent in G if and only if their distance in T is at most k. Solving a problem posed by Niedermeier [Personal communication, May 2004] we give a structural characterization of the graphs that have a 4-leaf root. Furthermore, we show that the graphs that have a 3-leaf root are essentially the trees, which simplifies a characterization due to Dom et al. [Error compensation in leaf power problems, Algorithmica 44 (2006) 363–381. (A preliminary version appeared under the title “Error compensation in leaf root problems”, in: Proceedings of the 15th Annual International Symposium on Algorithms and Computation (ISAAC 2004), Lecture Notes in Computer Science, vol. 3341, pp. 389–401)] and also a related recognition algorithm due to Nishimura et al. [On graph powers for leaf-labeled trees, J. Algorithms 42 (2002) 69–108].