Structure and linear-time recognition of 4-leaf powers

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 a k-leaf root of G. This notion was introduced and studied by Nishimura, Ragde, and Thilikos [2002], motivated by the search for underlying phylogenetic trees. Their results imply an O(n3 )-time recognition algorithm for 4-leaf powers. Recently, Rautenbach [2006] as well as Dom et al. [2005] characterized 4-leaf powers without true twins in terms of forbidden subgraphs. We give new characterizations for 4-leaf powers and squares of trees by a complete structural analysis. As a consequence, we obtain a conceptually simple linear-time recognition of 4-leaf powers. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems; G.2.2 [Discrete Mathematics]: Graph Theory General Terms: Algorithms, Theory Additional Key Words and Phrases: Graph powers, lea...

• Type: Journal
• Year: 2008
• Where: TALG
• Authors: Andreas Brandstädt, Van Bang Le, R. Sritharan
• Keywords: 4-leaf Powers, Algorithms, Phylogenetic Trees, TALG 2008