Boxicity of Leaf Powers Original Paper

First Online: 24 July 2010 Received: 09 March 2009 Revised: 22 June 2010 DOI :
10.1007/s00373-010-0962-5

Cite this article as: Chandran, L.S., Francis, M.C. & Mathew, R. Graphs and Combinatorics (2011) 27: 61. doi:10.1007/s00373-010-0962-5
Abstract The boxicity of a graph G , denoted as boxi(G ), is defined as the minimum integer t such that G is an intersection graph of axis-parallel t -dimensional boxes. A graph G is a k -leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k . Leaf powers are used in the construction of phylogenetic trees in evolutionary biology and have been studied in many recent papers. We show that for a k -leaf power G , boxi(G ) ≤ k −1. We also show the tightness of this bound by constructing a k -leaf power with boxicity equal to k −1. This result implies that there exist strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.

Keywords Boxicity Leaf powers Tree powers Strongly chordal graphs Interval graphs

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