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All polyhedral graphs with at most 16 vertices are Schurian: a computational verification
Li, Ponomarenko and Zeman state their belief that every polyhedral graph (i.e. every $3$-connected planar graph) is Schurian: that its coherent closure $WL(X)$ coincides with the orbital configuration of $Aut(X)$. Combined with their Theorem 1.1, this would imply that every polyhedral graph has Weis
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