On k- Versus (k + 1)-Leaf Powers

• Authors: Andreas Brandstädt;Peter Wagner
• Venue: COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
• Year: 2008

For k驴 2 and a finite simple undirected graph G= (V,E), a tree Tis a k-leaf rootof Gif Vis the set of leaves of Tand, for any two distinct x,y驴 V, xy驴 Eif and only if the distance between xand yin Tis at most k. Gis a k-leaf powerif Ghas a k-leaf root. Motivated by the search for underlying phylogenetic trees, the concept of k-leaf power was introduced and studied by Nishimura, Ragde and Thilikos and analysed further in many subsequent papers. It is easy to see that for all k驴 2, every k-leaf power is a (k+ 2)-leaf power. However, it was unknown whether every k-leaf power is a (k+ 1)-leaf power. Recently, Fellows, Meister, Rosamond, Sritharan and Telle settled this question by giving an example of a 4-leaf power which is not a 5-leaf power. Motivated by this result, we analyse the inclusion-comparability of k-leaf power classes and show that, for all k驴 4, the k- and (k+ 1)-leaf power classes are incomparable. We also characterise those graphs which are simultaneously 4- and 5-leaf powers.In the forthcoming full version of this paper, we will show that for all k驴 6 and odd lwith 3 ≤ l≤ k驴 3, the k- and (k+ l)-leaf power classes are incomparable. This settles all remaining cases and thus gives the complete inclusion-comparability of k-leaf power classes.