arXiv 2105.004362021-05cs.FL

由正则语言指定的图的性质

Properties of Graphs Specified by a Regular Language

Volker Diekert, Henning Fernau, Petra Wolf

备注：: 25 pages

摘要：

传统上，图算法得到一个单一的图作为输入，然后他们应该决定这个图是否满足一定的性质$Φ$。如果这个问题以一种方式修改，我们得到一个可能无限的图族作为输入，问题是在这个族中是否有一个图满足$Φ$？我们处理这个问题，通过使用正式的语言指定家庭的图形，特别是由经常的话。我们表明，某些图形属性可以决定通过研究规约语言$L$的语法幺半群，如果满足一定的扭转条件。如果$L$是正则的，则该条件通常成立。更具体地说，我们使用一个自然的二进制编码的有限图的二进制字母表$Σ$，我们定义了一个定期的集合$\mathbb{G}\subseteq Σ^*$，使每一个非空字$w\in \mathbb{G}$定义一个有限的和非空的图形。此外，图形属性可以在语法上定义为$Σ$上的语言。然后，我们问自动机$\mathcal{A}$是否指定了某个图满足一定的性质~ $Φ$。我们的结构结果表明，我们可以回答这个问题的所有“典型”的图形属性。为了说明我们的结果，我们将$L$分裂成一个有限的子集和这个联盟的每个子集定义在一个自然的方式，一个单一的有限图$F$的一些边缘和顶点被标记。标记图又定义了一个无限图$F^\infty$，因此定义了$F^\infty$的有限子图族，其中$F$作为导出子图出现。这产生了一个几何描述的所有图形指定的$L$的基础上分裂$L$成许多片，然后使用的概念，图收缩，我们得到一个容易理解的描述，在每一块的图形。

英文摘要：

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $Φ$. What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying $Φ$ in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language $L$ if a certain torsion condition is satisfied. This condition holds trivially if $L$ is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet $Σ$, and we define a regular set $\mathbb{G}\subseteq Σ^*$ such that every nonempty word $w\in \mathbb{G}$ defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over $Σ$. Then, we ask whether the automaton $\mathcal{A}$ specifies some graph satisfying a certain property~$Φ$. Our structural results show that we can answer this question for all "typical" graph properties. In order to show our results, we split $L$ into a finite union of subsets and every subset of this union defines in a natural way a single finite graph $F$ where some edges and vertices are marked. The marked graph in turn defines an infinite graph $F^\infty$ and therefore the family of finite subgraphs of $F^\infty$ where $F$ appears as an induced subgraph. This yields a geometric description of all graphs specified by $L$ based on splitting $L$ into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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