It can be shown that each permutation group $G \sqsubseteq S_n$ can be embedded, in a well defined sense, in a connected graph with $O(n+|G|)$ vertices. Some groups, however, require much fewer vertices. For instance, $S_n$ itself can be embedded in the $n$-clique $K_n$, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group $G \sqsubseteq S_n$ can be upper bounded by three structural parameters of connected graphs embedding $G$: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group $G \sqsubseteq S_n$ that can be embedded into a connected graph with $m$ vertices, treewidth k, and maximum degree $\Delta$, can also be generated by a context-free grammar of size $2^{O(k\Delta\log\Delta)}\cdot m^{O(k)}$. By combining our upper bound with a connection between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity $2^{O(k \Delta\log \Delta)}\cdot m^{O(k)}$. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated $2^{\Omega(n)}$ lower bound on the grammar complexity of the symmetric group $S_n$ we have that connected graphs of treewidth $o(n/\log n)$ and maximum degree $o(n/\log n)$ embedding subgroups of $S_n$ of index $2^{cn}$ for some small constant $c$ must have $n^{\omega(1)}$ vertices. This lower bound can be improved to exponential on graphs of treewidth $n^{\varepsilon}$ for $\varepsilon<1$ and maximum degree $o(n/\log n)$.

中文翻译：

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将置换组压缩为语法和多面体。图嵌入方法

可以证明，每个置换组 $G \sqsubseteq S_n$ 可以在定义明确的意义上嵌入到具有 $O(n+|G|)$ 顶点的连通图中。但是，某些组需要的顶点要少得多。例如，$S_n$ 本身可以嵌入到 $n$-clique $K_n$ 中，这是一个具有 n 个顶点的连通图。在这项工作中，我们证明了生成有限置换群 $G \sqsubseteq S_n$ 的上下文无关文法的最小尺寸可以由嵌入 $G$ 的连通图的三个结构参数确定上限：顶点数、树宽，以及最大程度。更准确地说，我们证明了任何可以嵌入到具有 $m$ 个顶点、树宽 k 和最大度数 $\Delta$ 的连通图中的置换群 $G \sqsubseteq S_n$，也可以由上下文无关文法生成大小为 $2^{O(k\Delta\log\Delta)}\cdot m^{O(k)}$。通过将我们的上限与置换群的扩展复杂度和形式语言的语法复杂度之间的联系相结合，我们还得到这些置换群可以用扩展复杂度 $2^{O(k \Delta\log \Delta)}\cdot m^{O(k)}$。上述上限可用于在置换组的索引与嵌入这些组的顶点数、树宽和连通图的最大度数之间进行权衡。特别是，通过将我们的主要结果与对称群 $S_n$ 的语法复杂性的著名 $2^{\Omega(n)}$ 下界相结合，我们得到了树宽 $o(n/\log n) 的连通图$ 和最大度数 $o(n/\log n)$ 嵌入索引 $2^{cn}$ 的 $S_n$ 的子群对于一些小的常数 $c$ 必须有 $n^{\omega(1)}$ 顶点。