Proof Path Graphs

rules = Flatten[With[{strings = Flatten[Table[StringJoin[#] & /@ Tuples[{"A", "B"}, i], {i, 1, 3}]]},Table[strings[[i]] <-> strings[[j]], {i, 1, Length[strings]}, {j,1, Length[strings]}]]]

h=[\[FilledSmallSquare]]	TwoWayStringTokenEventGraph [{rules[[2]]},3,"Sorting"->True,];
vertlist=DeleteCases[DeleteCases[#,_Integer]&/@VertexList[h],x_/;ResourceFunction["ContainsQ"][x,"Event"]];Table[Module[{h=[\[FilledSmallSquare]]	TwoWayStringTokenEventGraph [{rules[[2]]},3,"Sorting"->True,],proofs},
proofs=[\[FilledSmallSquare]]	LiteralizeTwoWayRules /@[\[FilledSmallSquare]]	FindTokenEventProof [h,{rules[[2]]},vertlist[[i]],8];
proofgraphs=Graph[#]&/@proofs; HighlightGraph[GraphUnion@@proofgraphs,proofgraphs]],{i,2,Length[vertlist]}]

j=[\[FilledSmallSquare]]	TwoWayStringTokenEventGraph [{rules[[110]]},3,"Sorting"->True,];
vertlist=DeleteCases[DeleteCases[#,_Integer]&/@VertexList[j],x_/;ResourceFunction["ContainsQ"][x,"Event"]];Table[Module[{h=[\[FilledSmallSquare]]	TwoWayStringTokenEventGraph [{rules[[110]]},3,"Sorting"->True,],proofs},
proofs=[\[FilledSmallSquare]]	LiteralizeTwoWayRules /@[\[FilledSmallSquare]]	FindTokenEventProof [h,{rules[[110]]},vertlist[[i]],8];
proofgraphs=Graph[#]&/@proofs;
[\[FilledSmallSquare]]	BranchialGraph [SimpleGraph[HighlightGraph[GraphUnion@@proofgraphs,proofgraphs]]]],{i,2,Length[vertlist]}]