Superfluous Arcs and Confluent Reductions in the Minimum Feedback Vertex Set Problem

Given a directed graph (digraph) G with vertex set V, a Feedback Vertex Set (FVS) is a subset of vertices whose removal eliminates all circuits in G. Finding a minimum feedback vertex set (MFVS) is NP-hard, but digraph reductions can reduce graph size while preserving at least one MFVS. This raises questions about the ordering in which reductions are applied and the existence of an optimal order that maximizes size reduction. The Church-Rosser property (confluence) ensures reductions can be applied in any order, leading to a unique reduced digraph up to isomorphism. In this work, we focus on arc reduction and its confluence within a broader set of known confluent reductions. We introduce Superfluous Arcs, which can be removed without affecting MFVS solutions, and propose a new parametrized reduction, <ns0:sc>chord</ns0:sc>k, to identify and remove specific superfluous arcs in polynomial time for bounded integer k. We establish the confluence of a set of reductions that includes <ns0:sc>chord</ns0:sc>k, creating the largest known confluent reduction system for MFVS, which improves preprocessing techniques for solving the MFVS problem efficiently.