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We present a definition of the class NP in combinatorial context as the setof languages of structures defined by finitely many forbidden liftedsubstructures. We apply this to special syntactically defined subclasses andshow how they correspond to naturally defined (and intensively studied)combinatorial problems. We show that some types of combinatorial problems likeedge colorings and graph decompositions express the full computational power ofthe class NP. We then characterize Constraint Satisfaction Problems (i.e.H-coloring problems) which are expressible by finitely many forbidden liftedsubstructures. This greatly simplifies and generalizes the earlier attempts tocharacterize this problem. As a corollary of this approach we perhaps find aproper setting of Feder and Vardi analysis of CSP languages within the classMMSNP.

Forbidden lists (NP and CSP for combinatorialists)