On characterizations by nice forbidding sets

Abstract Characterizations by forbidden subgraphs and forbidden minors are common in Graph Theory. In fact, any closed set in a qoset can have such a characterization, but in the general case these can only be guaranteed by forbidding the complement of the set, and do not seem to lend themselves to any practical applications. In this work, we begin a study of characterizations by forbidding sets which have nice properties, such as being minimal, finite, etc. More specifically, we characterize the qosets in which every closed set is characterizable by forbidding a minimal set, and show that the problem of determining whether a closed set in a general qoset has a finite forbidding set is undecidable. As a corollary, this problem, when restricted to the poset of finite graphs with the induced subgraph relation, is also undecidable.