Is there a class of finite graphs that can be axiomatized by existential second-order formulas but it's complement couldn't? (edited)

Question

I'm interested in the question above. I've just the add the "finite" requirement.

Answer 1

EDIT: This addresses the original version of the question, which did not include the finiteness stipulation. The current version of the question is equivalent to $\mathsf{NP}=\mathsf{coNP}$ (as mentioned by Emil Jerabke at MO), and so in particular is wildly open at present.

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The class of ill-founded graphs is the canonical example. A graph $G=(V,R)$ is ill-founded iff there is some subset $A\subseteq V$ such that for each $a\in A$ there is some $b\in A$ with $aRb$. This is obviously an existential second-order property, and the fact that its negation is not so expressible is one of the basic result in descriptive set theory and computability theory: a specific instance of the more general fact that the projective hierarchy does not collapse.

A full proof of this result is a bit long for an MSE answer, but a computability-based argument can be found in the early sections of Sacks' Higher recursion theory (which is freely and legally available at the link). We can also whip up a purely model-theoretic argument: existential second-order sentences are preserved upwards by ultraproducts, but the ultraproduct of well-founded graphs need not be well-founded. See here.