Vadim Lozin

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

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 $\mathbb{R}^2$.

How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an $n$-vertex graph $G$ is called implicit if it assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency of two vertices is a function of their codes. A necessary condition for a hereditary class $X$ of graphs to admit an implicit representation is that $X$ has at most factorial speed of growth. This condition, however, is not sufficient, as was recently shown in [Hatami & Hatami, FOCS 2022]. Several sufficient conditions for the existence of implicit representations deal with boundedness of some parameters, such as degeneracy or clique-width. In the present paper, we analyze more graph parameters and prove a number of new results related to implicit representation and factorial properties.