Polygonal methods permit easy meshing, even with elements of different shapes. Adaptive mesh refinement is crucial for achieving efficient approximations, particularly in the presence of singular solutions. Recently, residual based a posteriori error analyses have been developed for polygonal methods, including the virtual element method (VEM). This project will explore alternative a posteriori error estimation techniques to improve the efficiency constants, with a focus on the two-energies principle for the VEM. We will investigate equilibration techniques for second- and fourth-order problems, evaluating their robustness with respect to critical material parameters.