On (k,ℓ)-Leaf Powers

Abstract

We say that, for k ≥ 2 and ℓ> k, a tree T is a (k,ℓ)-leaf root of a graph G = (V _G ,E _G ) if V _G is the set of leaves of T, for all edges xy ∈ E _G , the distance d _T (x,y) in T is at most k and, for all non-edges $xy \not\in E_G$, d _T (x,y) is at least ℓ. A graph G is a (k,ℓ)-leaf power if it has a (k,ℓ)-leaf root. This new notion modifies the concept of k-leaf power which was introduced and studied by Nishimura, Ragde and Thilikos motivated by the search for underlying phylogenetic trees. Recently, a lot of work has been done on k-leaf powers and roots as well as on their variants phylogenetic roots and Steiner roots. For k = 3 and k = 4, structural characterisations and linear time recognition algorithms of k-leaf powers are known, and, recently, a polynomial time recognition of 5-leaf powers was given. For larger k, the recognition problem is open.

We give structural characterisations of (k,ℓ)-leaf powers, for some k and ℓ, which also imply an efficient recognition of these classes, and in this way we also improve and extend a recent paper by Kennedy, Lin and Yan on strictly chordal graphs and leaf powers.