On (k.l)-leaf powers
Seminar Room 1, Newton Institute
We say that, for k>1 and l>k, a tree T is a (k,l)-leaf root of a graph G=(V,E), if V is the set of leaves of T, for all edges xy in E, the distance d(x,y) between x and y in T (i.e., the number of edges of the unique path between the leaves x and y in T) is at most k and, for all non-edges xy outside E, d(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 generalises the concept of k-leaf power, which was introduced and studied by Nishimura, Ragde and Thilikos in 2002, motivated by the search for underlying phylogenetic trees: "...a fundamental problem in computational biology is the reconstruction of the phylogeny, or evolutionary history, of a set of species or genes, typically represented as a phylogenetic tree..." The species occur as leaves of the phylogenetic tree.
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 efficient recognition of these classes, and in this way we also improve and extend a recent paper from 2006 by Kennedy, Lin and Yan on strictly chordal graphs and leaf powers.
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