vines={{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}}],DualData[{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}}],DualData[{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}}],DualData[{1},{1},{1,2,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}}],DualData[{1},{1},{1,2,3}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}}],DualData[{1},{1},{3,2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}}],DualData[{1},{1},{3,2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{4,2,3,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}}],DualData[{1},{1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}}],DualData[{1},{1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}}],DualData[{1},{1},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{1,0,0}}],DualData[{1},{1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{1,0,0}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{1,0,0}}],DualData[{1},{1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{1,0,0}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{1,0,0}}],DualData[{1},{1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{1,0,0}}],DualData[{1},{1},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,1}}],DualData[{1},{1},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,0,1}}],DualData[{1},{1},{3,2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,0,1}}],DualData[{1},{1},{3,2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}}],DualData[{1},{1},{1,2,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}}],DualData[{1},{1},{1,2,3}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0},{0,1}}],DualData[{1},{1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0},{0,1}}],DualData[{1},{1},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1},{1,0}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1},{1,0}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{0,1}},{{1,0,0},{0,1,0},{0,0,1},{0,0,1}}],DualData[{1},{1},{1,2},{4,3,2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{1,0}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{0,1}},{{0,1,0},{0,0,1}},{{0,1}}],DualData[{1},{1},{3,2,1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{0,1}},{{0,0,1},{0,1,0}},{{1,0}}],DualData[{1},{1},{3,2,1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{2,1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{2,1},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}}],DualData[{1},{1},{1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}}],DualData[{1},{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0},{0,1}},{{1,0,0}}],DualData[{1},{1},{1,2},{1,2,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1},{1,0}},{{0,1}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{0,1}},{{1,0,0},{0,0,1}},{{0,1}}],DualData[{1},{1},{1,2},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1},{1,0}},{{1,0,0},{0,1,0}},{{0,1}}],DualData[{1},{1},{1,2},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}}],DualData[{1},{1},{1},{1,2,3},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}}],DualData[{1},{1},{1},{1,2,3},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}},{{1}}],DualData[{1},{1},{1},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}},{{1}}],DualData[{1},{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1}},{{1}},{{1}}],DualData[{1},{1},{1,2},{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1}},{{0,0,1}},{{1}}],DualData[{1},{1},{1,2},{3,2,1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0}},{{1}}],DualData[{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1}},{{0,1,0},{0,1,0},{0,0,1}},{{1,0,0},{0,1,0},{0,0,1},{0,0,1},{0,0,1}}],DualData[{1},{1},{1,2},{3,2,1},{5,4,3,2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0}},{{1},{1},{1}}],DualData[{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{1,0},{0,1}},{{0,0,0,1},{0,0,0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{4,2,3,1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{1,0},{0,1}},{{0,0,0,1},{0,0,0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{4,2,3,1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{1,0},{0,1}},{{0,0,0,1},{0,0,0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{4,3,2,1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{1,0},{0,1}},{{0,0,0,1},{0,0,0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{4,3,2,1},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{1,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,0,1},{1,0,0,0}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{4,2,3,1},{1,2,3,4}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,0,1},{1,0,0,0}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{4,2,3,1},{2,1,3,4}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,0,1},{1,0,0,0}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{4,2,3,1},{2,1,4,3}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{1,2},{4,3,2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,0,1},{1,0,0,0}},{{1,1}}],DualData[{1},{1},{1,2},{4,2,3,1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}},{{1}}],DualData[{1},{1},{1},{1,2,3},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}},{{1}}],DualData[{1},{1},{1},{1,2,3},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}},{{1}},{{1}}],DualData[{1},{1},{1},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1}},{{1},{1},{1}},{{1,0,0},{0,1,0}},{{1,0}},{{1}},{{1}}],DualData[{1},{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,0,1},{1,0,0,0}},{{1,1}},{{1}},{{1}}],DualData[{1},{1},{1,2},{4,2,3,1},{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{1,0,0,1}}],DualData[{1},{1},{1,2},{1,2},{4,3,2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,1}},{{1}},{{1}}],DualData[{1},{1},{1,2},{1,2},{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,0,1},{1,0,0,0}},{{1,0},{1,0},{0,1},{0,1}},{{1,0,1,0}}],DualData[{1},{1},{1,2},{4,2,3,1},{1,2,3,4}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0},{0,1}},{{1,0,0},{0,1,0},{0,0,1}},{{0,1,0},{0,0,1}},{{1,0},{0,1}},{{0,1}},{{1}}],DualData[{1},{1},{1,2},{3,2,1},{2,1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1},{1,0}},{{1,0,0},{0,1,0},{0,1,0}},{{1,0,0},{0,0,1}},{{1,0}},{{1}}],DualData[{1},{1},{1,2},{3,2,1},{1}]]}};

hexagon=vines[[17]]

FindFusionRules[hexagon[[1]],hexagon[[2]]]

DisplayBigraph[vines[[16]]]

Clear[ffr]
ffr[n_,k_]:=ffr[n,k]=(Clear[Z];FindFusionRules[Translate[vines[[n,1]],k],Translate[vines[[n,2]],k]])

SetDebugMode[False];SetDebugQueueInterval[1.];

ffr[10,0]

Do[Print[n," ",DisplayBigraph[vines[[n]]]," ",Length[ffr[n,0]]],{n,1,Length[vines]}]

Cases[Range[30],n_/;Length[ffr[n,0]]==0]

{h1,h2}=vines[[16]]

FindFusionRules[h1,h2]

Clear[Z]
FindPartialFusionRules[h1,h2,Z,True]

{bs1,bs2}={BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}}],DualData[{1},{1},{1,2},{1,2},{2,1}]]};

Clear[Z]
FindPartialFusionRules[bs1,bs2,Z,True]

SetDebugMode[True]

Clear[Z]
FindPartialFusionRules[Translate[bs1,4],Translate[bs2,4],Z,True]

Union[Cases[%,_Z,∞]]

Clear[Z]
FindPartialFusionRules[Translate[bs1,8],Translate[bs2,8],Z,True]

Now try one of the extensions of the bad seed.

{bse1,bse2}={BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2},{2,1},{1,2},{2,1},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2},{2,1},{1,2},{2,1}]]};

DisplayBigraph/@{bse1,bse2}

GraphDepth/@{bs1,bse1}

SetDebugMode[True]

Clear[Z]
FindPartialFusionRules[Translate[Truncate[bse1,-4],4],Translate[Truncate[bse2,-4],4],Z,True]

This one would probably complete in large but finite time :

Clear[Z]
FindPartialFusionRules[Translate[Truncate[bse1,-2],4],Translate[Truncate[bse2,-2],4],Z,True]

And this one takes too much memory :

Clear[Z]
FindPartialFusionRules[Translate[bse1,4],Translate[bse2,4],Z,True]