In[]:=
PlanarTest[rule_,t_Integer:1]:=LengthWhile[$RandomGraphList,PlanarGraphQ[Echo@Graph[UndirectedEdge@@@WolframModel[rule,#,t,"FinalState"]]]&]Length[$RandomGraphList]
In[]:=
With[{ru=EnumerateWolframModelRules[{{1,2}}{{2,2}}]},Pick[ru,ParallelMapMonitored[PlanarTest[#]&,ru]]]
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Local machine only
Local machine only