Feedback Vertex Set (FVS)

The Feedback Vertex Set (FVS) problem in graph theory and computer science involves finding the smallest subset of vertices in an undirected graph such that removing these vertices results in a graph with no cycles, i.e., a forest. The goal of the FVS problem is to minimize the size of the feedback vertex set, and this problem is NP-hard, meaning that finding an optimal solution has high computational complexity. For directed graphs, a feedback vertex set refers to a subset of vertices whose removal results in a directed acyclic graph (DAG). The FVS problem has significant application value in optimizing network structures and reducing system cyclic dependencies.