In this paper we give a historical and theoretical background to minimal triangulation and its relation to minimal separators.
We introduce a new type of minimal separators, the minimal meta separator, which its size is a lower bound for the treewidth problem, its fill-in is a lower bound for the minimum fill-in problem and can be used for the aid of enumeration of minimal separators.
Furthermore, we introduce the idea of transformation which are controlled graph modifications that predictably modify the minimal separators of such graph. Namely, we introduce the transformations of inclusion minimal clique decomposition, saturate minimal separator, edge addition and minimal meta separator decomposition. These transformations can aid the architecting of algorithms for the enumeration of minimal separators.
Finally, we then apply such methods on the problem of enumeration of minimal almost-clique separators and demonstrate faster runtimes compared to the current state of the art.