Algebraic Characterizations for Minors of Finite Graphs via Flow Transformation Monoid Division and Embedding

2026-08-25

AI Summary

该研究通过流变换幺半群的划分和嵌入，解决了有限图子图的代数刻画问题。

Abstract

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.

Problem

• flow monoid
• graph minors
• algebraic characterization

Innovation

• flow transformation monoid
• graph minors
• algebraic characterization
• embedded-copy theorem