A partial automorphism of a finite graph is an isomorphism between its vertex-induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of standard tools of inverse semigroup theory, namely Green’s relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.

中文翻译：

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部分图自同构的逆单半体

有限图的部分自同构是其顶点诱导的子图之间的同构。给定有限图的所有部分自同构的集合在（部分图的）组成下形成逆单等分。我们通过逆半群论的标准工具，即格林的关系和自然偏序的某些性质，描述了这种逆半群的代数结构，并给出了作为部分图自同构的逆半群出现的逆半群的特征。我们也将结果扩展到有向图和边色有向图。