Characterising (k,l)-leaf powers

  • Authors:

  • Andreas Brandstädt;Peter Wagner

  • Affiliations:

  • Institut für Informatik, Universität Rostock, D-18051 Rostock, Germany;Institut für Informatik, Universität Rostock, D-18051 Rostock, Germany

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

We say that, for k=2 and @?k, a tree T with distance function d"T(x,y) is a (k,@?)-leaf root of a finite simple graphG=(V,E) if V is the set of leaves of T, for all edges xy@?E, d"T(x,y)@?k, and for all non-edges xy@?E, d"T(x,y)=@?. A graph is a (k,@?)-leaf power if it has a (k,@?)-leaf root. This new notion modifies the concept of k-leaf powers (which are, in our terminology, the (k,k+1)-leaf powers) introduced and studied by Nishimura, Ragde and Thilikos; k-leaf powers are 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. Many problems, however, remain open. We give the structural characterisations of (k,@?)-leaf powers, for some k and @?, which also imply an efficient recognition of these classes, and in this way we improve and extend a recent paper by Kennedy, Lin and Yan on strictly chordal graphs; one of our motivations for studying (k,@?)-leaf powers is the fact that strictly chordal graphs are precisely the (4,6)-leaf powers.