Graph Layout Problems refer to a family of optimization problems where the aim is to assign the vertices of an input graph to the vertices of a structured host graph, optimizing a certain objective function. In this paper, we tackle one of these problems, named Cyclic Antibandwidth Problem, where the objective is to maximize the minimum distance of all adjacent vertices, computed in a cycle host graph. Specifically, we propose a General Variable Neighborhood Search which combines an efficient Variable Neighborhood Descent with a novel destruction–reconstruction shaking procedure. Additionally, our proposal takes advantage of two new exploration strategies for this problem: a criterion for breaking the tie of solutions with the same objective function and an efficient evaluation of neighboring solutions. Furthermore, two new neighborhood reduction strategies are proposed. We conduct a thorough computational experience by comparing the algorithm proposed with the current state-of-the-art methods over a set of previously reported instances. The associated results show the merit of the introduced algorithm, emerging as the best performance method in those instances where the optima are unknown. These results are further confirmed with nonparametric statistical tests.