The main goal of this paper is to bring a contribution in order to facilitate automatic refinement of Bimatrix Game Nash extreme equilibria. We show how maximal cliques enumeration algorithms can be used to enumerate all Nash maximal subsets of a Bimatrix Game. We also use a pair of linear programs to identify perfect extreme equilibria and enumerate all Selten Maximal Subsets. These two results are used by the XGame-Solver software to automatically generate all Selten Maximal Subsets corresponding to a Bimatrix game.