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Forbidden subgraphs in terms of forbidden quantifiers

In 1930, Kuratowski characterized planar graphs as those graphs which fail to contain either of two special subgraphs; see Theorem 11.13 of [4]. Since then, such ''forbidden subgraph" characterizations have been sought and prized by graph theorists. The nature of such characterizations is considered in [2] and [3]. In particular, [3] is based on the simple observation that a class of graphs has a forbidden subgraph characterization if and only if the class contains each subgraph of each of its members. We will show that the properties characterizable using forbidden subgraphs are precisely those which are expressible in a natural symbolic language from which existential quantifiers have been forbidden. Of course, this is exactly what is expected from combining the observation of [3] with the well known result of Tarski and Los on properties preserved under subsystems (Theorem U of [5]). But unlike either of these approaches, ours uses only very simple symbolic logic and is actually able to produce the set of forbidden subgraphs.