Variable Neighborhood Search (VNS) is a metaheuristic for solving optimization problems based on a systematic change of neighborhoods. In recent years, a large variety of VNS strategies have been proposed. However, we have only found limited experimental comparisons among different VNS variants. This paper reviews three VNS strategies for finding near-optimal solutions for vertex-cut minimization problems. Specifically, we consider the min-max variant (Vertex Separation Problem) and the min-sum variant (SumCut Minimization Problem). We also present an preliminary computational comparison of the methods on previously reported instances.