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In this paper, we use skein-theoretic techniques to classify all virtual knotpolynomials and trivalent graph invariants with certain smallness conditions.The first half of the paper classifies all virtual knot polynomials givingnon-trivial invariants strictly smaller than the one given by the Higman-Simsspin model. In particular, we exhibit a family of skein theories coming from$\text{Rep}(O(2))$ with an interesting braiding. In addition, all skeintheories of oriented virtual tangles with some smallness conditions areclassified. In the second half of the paper, we classify all non-trivialinvariants of (perhaps non-planar) trivalent graphs coming from symmetrictrivalent categories. For each of these categories, we also classify when thesub-category generated by only the trivalent vertex is braided. An interestingexample of this arise from the tensor category Deligne's $S_t$.

Skein theories for virtual tangles