Ptolemaic Graphs and Interval Graphs Are Leaf Powers

  • Andreas Brandstädt
  • Christian Hundt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)


Motivated by the problem of reconstructing evolutionary history, Nishimura, Radge and Thilikos introduced the notion of k-leaf powers as the class of graphs G = (V,E) which have a k-leaf root, i.e., a tree T with leaf set V where xy ∈ E if and only if the T-distance between x and y is at most k. It is known that leaf powers are strongly chordal (i.e., sun-free chordal) graphs. Despite extensive research, the problem of recognizing leaf powers, i.e., to decide for a given graph G whether it is a k-leaf power for some k, remains open. Much less is known on the complexity of finding the leaf rank of G, i.e., to determine the minimum number k such that G is a k-leaf power. A result by Bibelnieks and Dearing implies that not every strongly chordal graph is a leaf power. Recently, Kennedy, Lin and Yan have shown that dart- and gem-free chordal graphs are 4-leaf powers. We generalize their result and show that ptolemaic (i.e., gem-free chordal) graphs are leaf powers. Moreover, ptolemaic graphs have unbounded leaf rank. Furthermore, we show that interval graphs are leaf powers which implies that leaf powers have unbounded clique-width. Finally, we characterize unit interval graphs as those leaf powers having a caterpillar leaf root.

Keywords and Classification

Leaf powers leaf roots strongly chordal graphs ptolemaic graphs graph powers graph class inclusions (unit) interval graphs clique-width 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andreas Brandstädt
    • 1
  • Christian Hundt
    • 1
  1. 1.Institut für Informatik, Universität Rostock, D-18051Germany

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