Local and Union Boxicity

TL;DR

This paper introduces local and union boxicity, new graph parameters related to intersection representations with axis-aligned boxes, and explores their properties and relationships to classical boxicity.

Contribution

It defines local and union boxicity, establishes inequalities among them and classical boxicity, and shows their characterizations via intersection representations, highlighting their potential to better indicate graph complexity.

Findings

Local and union boxicity are bounded by classical boxicity.

These parameters can differ arbitrarily far from each other.

Local boxicity may better reflect graph complexity.

Abstract

The boxicity $\operatorname{box}(H)$ of a graph $H$ is the smallest integer $d$ such that $H$ is the intersection of $d$ interval graphs, or equivalently, that $H$ is the intersection graph of axis-aligned boxes in $\mathbb{R}^d$. These intersection representations can be interpreted as covering representations of the complement $H^c$ of $H$ with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, Discrete Mathematics 339 (2016)) to define two new parameters: the local boxicity $\operatorname{box}_\ell(H)$ and the union boxicity $\overline{\operatorname{box}}(H)$ of $H$. The union boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with $d$ vertex-disjoint unions of co-interval graphs, while the local boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with co-interval graphs, at most $d$ at every vertex. We show that for every graph $H$ we have $\operatorname{box}_\ell(H) \leq \overline{\operatorname{box}}(H) \leq \operatorname{box}(H)$ and that each of these inequalities can be arbitrarily far apart. Moreover, we show that local and union boxicity are also characterized by intersection representations of appropriate axis-aligned boxes in $\mathbb{R}^d$. We demonstrate with a few striking examples, that in a sense, the local boxicity is a better indication for the complexity of a graph, than the classical boxicity.