Perfect couples of graphs

We generalize the concept of perfect graphs in terms of additivity of a functional called graph entropy. The latter is an information theoretic functional on a graphG with a probability distributionP on its vertex set. For any fixedP it is sub-additive with respect to graph union. The entropy of the complete graph equals the sum of those ofG and its complement G iffG is perfect. We generalize this recent result to characterize all the cases when the sub-additivity of graph entropy holds with equality.The research of the authors is partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant No. 1806 resp. No. 1812.