AI 中文总结
该研究通过流变换幺半群的除法与嵌入，给出有限图连通性、子式的代数刻画及嵌入拷贝定理，为图论子式问题提供了代数视角的新方法。
AI 中文摘要
我们证明了关于有限图流幺半群的三个定理。首先，非空有限图G=(V,E)是连通的，当且仅当它的流幺半群包含V上的一个常值映射，等价于它包含V上所有常值映射。其次，我们通过流变换幺半群的除法给出了图子式的一种新刻画，同时给出了一个代数交叉条件，用于检测待收缩顶点集之间的边。第三，我们将这一结论强化为嵌入拷贝定理：图M是G的子式，当且仅当在满足类似交叉条件的前提下，M的流变换幺半群可实现为G的环境流幺半群的一个子半群的诱导作用，该子半群是具有局部幂等元单位的幺半群。
英文摘要
We prove three theorems on the flow monoids of finite graphs. First, we show that a non-empty finite graph G = (V, E) is connected if and only if its flow monoid contains a constant map on V, equivalently, if and only if it contains all constant maps on V. Second, we give a new characterization of graph minors in terms of division of flow transformation monoids, together with an algebraic crossing condition that detects edges between the vertex sets being contracted. Third, we strengthen this to an embedded-copy theorem: a graph M is a minor of G if and only if, subject to analogous crossing conditions, the flow transformation monoid of M is realized as the induced action of a subsemigroup of the ambient flow monoid of G, this subsemigroup being a monoid with a local idempotent identity.
CommentsIn Proceedings AFL 2026, arXiv:2608.23071
Journal refEPTCS 451, 2026, pp. 48-59
