  [Alecu et al.: Graph functionality, JCTB2021] define functionality, a graph
parameter that generalizes graph degeneracy. They research the relation of
functionality to many other graph parameters (tree-width, clique-width,
VC-dimension, etc.). Extending their research, we prove logarithmic lower bound
for functionality of random graph $G(n,p)$ for large range of $p$. Previously
known graphs have functionality logarithmic in number of vertices. We show that
for every graph $G$ on $n$ vertices we have $\mathrm{fun}(G) \le O(\sqrt{ n
\log n})$ and we give a nearly matching $\Omega(\sqrt{n})$-lower bound provided
by projective planes.
  Further, we study a related graph parameter \emph{symmetric difference}, the
minimum of $|N(u) \Delta N(v)|$ over all pairs of vertices of the ``worst
possible'' induced subgraph. It was observed by Alecu et al. that
$\mathrm{fun}(G) \le \mathrm{sd}(G)+1$ for every graph $G$. We compare
$\mathrm{fun}$ and $\mathrm{sd}$ for the class $\mathrm{INT}$ of interval
graphs and $\mathrm{CA}$ of circular-arc graphs. We let $\mathrm{INT}_n$ denote
the $n$-vertex interval graphs, similarly for $\mathrm{CA}_n$. Alecu et al.
ask, whether $\mathrm{fun}(\mathrm{INT})$ is bounded. Dallard et al. answer
this positively in a recent preprint. On the other hand, we show that
$\Omega(\sqrt[4]{n}) \leq \mathrm{sd}(\mathrm{INT}_n) \leq O(\sqrt[3]{n})$. For
the related class $\mathrm{CA}$ we show that $\mathrm{sd}(\mathrm{CA}_n) =
\Theta(\sqrt{n})$. We propose a follow-up question: is
$\mathrm{fun}(\mathrm{CA})$ bounded?
