Boxicity of Leaf Powers

TL;DR

This paper establishes an upper bound on the boxicity of k-leaf powers, showing it is at most k-1, and demonstrates the bound's tightness, revealing that strongly chordal graphs can have arbitrarily high boxicity.

Contribution

The paper proves that the boxicity of k-leaf powers is at most k-1 and constructs examples showing this bound is tight, advancing understanding of graph complexity.

Findings

Box(G) q k-1 for k-leaf powers

Constructed k-leaf powers with boxicity exactly k-1

Strongly chordal graphs can have arbitrarily high boxicity

Abstract

The boxicity of a graph G, denoted as box(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 a subclass of strongly chordal graphs and are used in the construction of phylogenetic trees in evolutionary biology. We show that for a k-leaf power G, box(G)\leq 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 exists strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.