In graph theory, the minimum directed feedback vertex set (FVS) problem consists in identifying the smallest subsets of vertices in a directed graph whose deletion renders the directed graph acyclic. Although being known as NP-hard since 1972, this problem can be solved in a reasonable time on small instances, or on instances having special combinatorial structure. In this paper we investigate graph reductions preserving all or some minimum FVS and focus on their properties, especially the Church-Rosser property, also called confluence. The Church-Rosser property implies the irrelevance of reduction order, leading to a unique directed graph. The study seeks the largest subset of reductions with the Church-Rosser property and explores the adaptability of reductions to meet this criterion. Addressing these questions is crucial, as it may impact algorithmic implications, allowing for parallelization and speeding up sequential algorithms.

翻译：在图论中，最小有向反馈顶点集问题旨在识别有向图中最小的顶点子集，删除这些顶点后可使有向图变为无环。尽管自1972年以来该问题已知为NP难问题，但在小规模实例或具有特殊组合结构的实例上，仍可在合理时间内求解。本文研究保持全部或部分最小反馈顶点集的图归约，并重点关注其性质，尤其是Church-Rosser性质（亦称合流性）。Church-Rosser性质意味着归约顺序的无关性，从而得到唯一的有向图。本研究旨在寻找具有Church-Rosser性质的最大归约子集，并探讨使归约适应此标准的可能性。解决这些问题至关重要，因为它可能影响算法设计，实现并行化并加速串行算法。