Functionality of Box Intersection Graphs

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 \(\mathbb {R}^1\) ${\mathbb{R}}^1$ , i.e. for interval graphs, and unbounded for box intersection graphs in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ . 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 \(\mathbb {R}^2\) ${\mathbb{R}}^2$ .