Functionality of box intersection graphs

TL;DR

This paper investigates the complexity measure called functionality in box intersection graphs, showing it is bounded in one dimension but unbounded in three, and explores related parameters like symmetric difference.

Contribution

It establishes bounds for functionality in box intersection graphs across different dimensions and analyzes the unbounded nature of symmetric difference in certain graph classes.

Findings

Functionality is bounded for interval graphs in $\

$ ext{R}^1$ but unbounded in $ ext{R}^3$.

Symmetric difference parameter is unbounded for interval and unit box intersection graphs in $ ext{R}^2$.

Abstract

Functionality is a graph complexity measure that extends a variety of parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In the present paper, we show that functionality is bounded for box intersection graphs in $\mathbb{R}^1$, i.e. for interval graphs, and unbounded for box intersection graphs in $\mathbb{R}^3$. We also study a parameter known as symmetric difference, which is intermediate between twin-width and functionality, and show that this parameter is unbounded both for interval graphs and for unit box intersection graphs in $\mathbb{R}^2$.