Perfect graphs form a well-known class of graphs introduced by Berge in the 1960s in terms of a min-max type equality involving two famous graph parameters. In this work, we study variants and subclasses of perfect graphs defined by means of min-max relations of other graph parameters. Our focus is on clique-perfect, coordinated, and neighborhoodperfect graphs. We show the connection between graph classes and both hypergraph theory and the clique graph operator. We review different partial characterizations of them by forbidden induced subgraphs, present the previous results, and the main open problems. Computational complexity problems are also discussed.