Boxicity of Leaf Powers
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.
KeywordsBoxicity Leaf powers Tree powers Strongly chordal graphs Interval graphs
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