Structure and linear time recognition of 3-leaf powers

  • Authors:

  • Andreas Brandstädt;Van Bang Le

  • Affiliations:

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

  • Venue:
  • Information Processing Letters
  • Year:
  • 2006

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Abstract

A graph G is the k-leaf power of a tree T if its vertices are leaves of T such that two vertices are adjacent in G if and only if their distance in T is at most k. Then T is the k-leaf root of G. This notion was introduced and studied by Nishimura, Ragde, and Thilikos motivated by the search for underlying phylogenetic trees. Their results imply a O(n^3) time recognition algorithm for 3-leaf powers. Later, Dom, Guo, Huffner, and Niedermeier characterized 3-leaf powers as the (bull,@?dart,@?gem)-free chordal graphs. We show that a connected graph is a 3-leaf power if and only if it results from substituting cliques into the vertices of a tree. This characterization is much simpler than the previous characterizations via critical cliques and forbidden induced subgraphs and also leads to linear time recognition of these graphs.