[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?

翻译：[Alecu等学者：图功能性，JCTB2021]定义了功能性这一图参数，它是图退化度的推广。他们研究了功能性与众多其他图参数（树宽、团宽、VC维等）之间的关系。扩展他们的研究，我们证明了随机图$G(n,p)$在$p$的大范围取值下功能性的对数下界。此前已知图的功能性关于顶点数呈对数增长。我们证明对任意$n$个顶点的图$G$有$\mathrm{fun}(G) \le O(\sqrt{ n \log n})$，并利用射影平面给出了一个几乎匹配的$\Omega(\sqrt{n})$下界。此外，我们研究了一个相关的图参数\emph{对称差}，即所有顶点对中“最差情况”诱导子图的$|N(u) \Delta N(v)|$最小值。Alecu等学者指出，对任意图$G$有$\mathrm{fun}(G) \le \mathrm{sd}(G)+1$。我们比较了区间图类$\mathrm{INT}$和圆弧图类$\mathrm{CA}$上的$\mathrm{fun}$与$\mathrm{sd}$。记$\mathrm{INT}_n$为$n$顶点区间图，$\mathrm{CA}_n$类似定义。Alecu等学者问$\mathrm{fun}(\mathrm{INT})$是否有界，Dallard等学者在近期的预印本中给出了肯定回答。另一方面，我们证明$\Omega(\sqrt[4]{n}) \leq \mathrm{sd}(\mathrm{INT}_n) \leq O(\sqrt[3]{n})$。对于相关类$\mathrm{CA}$，我们证明$\mathrm{sd}(\mathrm{CA}_n) = \Theta(\sqrt{n})$并提出后续问题：$\mathrm{fun}(\mathrm{CA})$是否有界？