1,2  3,2

GraphPlot[Rule@@@WolframModel[{{1,2}}{{2,1},{3,1},{3,2}},{{0,0}},5,"FinalState"]]
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GraphPlot[Rule@@@WolframModel[{{1,2}}{{2,1},{3,1},{3,2}},{{0,0}},7,"FinalState"]]
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GraphPlot[Rule@@@WolframModel[{{1,2}}{{2,3},{2,3},{3,1}},{{0,0}},7,"FinalState"]]
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GraphPlot3D[Rule@@@WolframModel[{{1,2}}{{2,3},{2,3},{3,1}},{{0,0}},7,"FinalState"]]
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GraphPlot3D[Rule@@@WolframModel[{{1,2}}{{2,3},{2,3},{3,1}},{{0,0}},3,"FinalState"]]
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1,2 4,2

Just 3 symbols:
EchoFunction[Length]@Select[EnumerateWolframModelRules[{{1,2}}{{4,2}},3],BiConnectedRuleQ];
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618
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res=ParallelMapMonitored[WolframModelTest[#,Automatic]&,%];
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MakePictures[Select[res,WMFilter4[#]==="LinearRecurrenceGrowth"&]]
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{4,-3}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{4,-3}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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{3,-2}
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NOTE: this search uses 10 instead of 6 max symbols
res=ParallelMapMonitored[RandomWolframModelTestConnected[{{{{1,2}}{{4,2}},10}}]&,Range[50]];
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MakePictures[res]
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Out[]=
res=ParallelMapMonitored[RandomWolframModelTestConnected[{{{{1,2}}{{4,2}},10}}]&,Range[50]];
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MakePictures[res]
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Out[]=
{{{{{1,2}}{{1,3},{3,1},{3,2},{4,3}}},{{0,0}},4},{{{{1,2}}{{1,3},{3,2},{3,4},{5,3}}},{{0,0}},4}}
GraphPlot[Rule@@@WolframModel[{{{1,2}}{{1,3},{3,1},{3,2},{4,3}}},{{0,0}},6,"FinalState"]]
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GraphPlot[Rule@@@WolframModel[{{{1,2}}{{1,3},{3,2},{3,4},{5,3}}},{{0,0}},6,"FinalState"]]
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res=ParallelMapMonitored[RandomWolframModelTestConnected[{{{{1,2}}{{4,2}},6}}]&,Range[50]];
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MakePictures[res]
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Out[]=
GraphPlot[Rule@@@(WolframModel@@#)["FinalState"]]&/@{{{{{1,2}}{{2,3},{2,4},{3,1},{3,4}}},{{0,0}},6},{{{{1,2}}{{1,3},{2,3},{2,3},{4,3}}},{{0,0}},6}}
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res=ParallelMapMonitored[RandomWolframModelTestConnected[{{{{1,2}}{{4,2}},6}}]&,Range[50]];
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MakePictures[res]
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Out[]=
GraphPlot[Rule@@@(WolframModel@@#)["FinalState"]]&/@{{{{{1,2}}{{1,3},{3,2},{4,1},{4,2}}},{{0,0}},6},{{{{1,2}}{{1,3},{2,1},{2,3},{3,2}}},{{0,0}},5}}
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1,32,3

EchoFunction[Length]@Select[EnumerateWolframModelRules[{{1,3}}{{2,3}},3],BiConnectedRuleQ];
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1613
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res=WolframModelTest[#,Automatic]&/@%;
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ReverseSort[Counts[WMFilter4/@res]]
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MakePictures[First/@Values[GroupBy[Select[res,WMFilter4[#]==="PureExponential"&],#FinalState&]]]
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