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You can run the Euclidean algorithm of $\mathbb{Q}$ and clear denominators to get a universal bound: there are integers p,q,r so that pf(x) +qg(x) =r and if you evaluate at $a$ the gcd f(a) and g(a) must divide $r$.
Perhaps the non-evasiveness has to be addressed further given that the proof changed. 1) the ones described in the problem. 2) the ones obtained from removing a vertex, which give an additional local condition to relax the larger one, i.e there is a vertex with a pleasant link! It seems like this imposes many conditions on the graph. Do you have a systematic way to construct interesting examples?
