On (k, l)-leaf powers

  • Authors:

  • Andreas Brandstädt;Peter Wagner

  • Affiliations:

  • Institut für Informatik, Universität Rostock, Rostock, Germany;Institut für Informatik, Universität Rostock, Rostock, Germany

  • Venue:
  • MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
  • Year:
  • 2007

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Abstract

We say that, for k ≥ 2 and l k, a tree T is a (k, l)-leaf root of a graph G = (VG,EG) if VG is the set of leaves of T, for all edges xy ∈ EG, the distance dT (x, y) in T is at most k and, for all nonedges xy ∈ EG dT (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 kleaf 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.