  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.
