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In this research we consider minimal hard and boundary classes. These classes are of considerable importance in demarcating the polynomial solvability and the  NP -completeness of problems on graphs in the family of hereditary classes. Earlier it was established that there are no minimal hard classes for a certain type of problems on graphs. Here we find a sufficient condition for existence of such classes. With its use we construct new examples of minimal hard classes and boundary classes for some problems on graphs. We provide complete description of the sets of minimal hard and boundary classes for the problems under consideration. Up to now, no such descriptions were known.

Extremal sets of graphs in the problem of demarcation in the family of hereditary closed classes of graphs