Get["https://raw.githubusercontent.com/szhorvat/IGraphM/master/IGInstaller.m"]

<<IGraphM`

IGDocumentation[]

Graph[GraphData["DodecahedralGraph"],EdgeStyle->Directive[Thick,Opacity[1]],VertexStyle->Black]//IGEdgeMap[ColorData[109],EdgeStyle->IGEdgeColoring]

g=IGTriangularLattice[3,VertexShapeFunction->"Name"]

IGFaces[g]
IGDualGraph[g]

PlanarGraph[IGDualGraph[g,VertexLabels->"Name"]]

adjv7f81=Normal[AdjacencyMatrix[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]4,1\[UndirectedEdge]4,5\[UndirectedEdge]6,6\[UndirectedEdge]7,5\[UndirectedEdge]7,1\[UndirectedEdge]5,1\[UndirectedEdge]7,4\[UndirectedEdge]7,3\[UndirectedEdge]6,3\[UndirectedEdge]7,2\[UndirectedEdge]6}]]

adjv7f81g=PlanarGraph[AdjacencyGraph[adjv7f81,VertexLabels->"Name"]]

AdjacencyGraph[adjv7f81,VertexLabels->"Name"]

IGFaces[adjv7f81g]

IGDualGraph[adjv7f81g,VertexLabels->"Name"]

adjv7f81dg=PlanarGraph[IGDualGraph[adjv7f81g,VertexLabels->"Name"]]

IGFaces[adjv7f81dg]

adjv7f81d=Normal[AdjacencyMatrix[adjv7f81dg]]

Table[If[i==j,1,If[adjv7f81[[i,j]]==1,-1,If[i>j,(Subscript[b, j,i]),(Subscript[b, i,j]),]]],{i,1,VertexCount[adjv7f81g]},{j,1,VertexCount[adjv7f81g]}]//MatrixForm

Table[If[i==j,1,If[adjv7f81d[[i,j]]==1,-1,If[i>j,(Subscript[b, j+VertexCount[adjv7f81g],i+VertexCount[adjv7f81g]]),(Subscript[b, i+VertexCount[adjv7f81g],j+VertexCount[adjv7f81g]]),]]],{i,1,VertexCount[adjv7f81dg]},{j,1,VertexCount[adjv7f81dg]}]//MatrixForm

Clear[i]

Table[If[MemberQ[IGFaces[adjv7f81g][[j]],i],0,(Subscript[b, j+VertexCount[adjv7f81g],i])],{i,VertexCount[adjv7f81g]},{j,VertexCount[adjv7f81dg]}]//MatrixForm

Table[If[MemberQ[IGFaces[adjv7f81g][[j]],i],0,If[i>j,(Subscript[b, i+VertexCount[adjv7f81g],j]),(Subscript[b, i+VertexCount[adjv7f81g],j])]],{i,1,VertexCount[adjv7f81g]},{j,1,VertexCount[adjv7f81dg]}]//MatrixForm

ArrayFlatten[{{Table[If[i==j,1,If[adjv7f81[[i,j]]==1,-1,If[i>j,(Subscript[b, j,i]),(Subscript[b, i,j])]]],{i,1,VertexCount[adjv7f81g]},{j,1,VertexCount[adjv7f81g]}],Table[If[MemberQ[IGFaces[adjv7f81g][[j]],i],0,(Subscript[b, j+VertexCount[adjv7f81g],i])],{i,VertexCount[adjv7f81g]},{j,VertexCount[adjv7f81dg]}]},{Transpose[Table[If[MemberQ[IGFaces[adjv7f81g][[j]],i],0,(Subscript[b, j+VertexCount[adjv7f81g],i])],{i,VertexCount[adjv7f81g]},{j,VertexCount[adjv7f81dg]}]],Table[If[i==j,1,If[adjv7f81d[[i,j]]==1,-1,If[i>j,(Subscript[b, j+VertexCount[adjv7f81g],i+VertexCount[adjv7f81g]]),(Subscript[b, i+VertexCount[adjv7f81g],j+VertexCount[adjv7f81g]])]]],{i,1,VertexCount[adjv7f81dg]},{j,1,VertexCount[adjv7f81dg]}]}}]//MatrixForm

Clear[adjmat]

pgraph[adjmat_]:=PlanarGraph[AdjacencyGraph[adjmat,VertexLabels->"Name"]]

dgraph[adjmat_]:=PlanarGraph[IGDualGraph[pgraph[adjmat],VertexLabels->"Name"]]

dadjmat[adjmat_]:=Normal[AdjacencyMatrix[dgraph[adjmat]]]

exGmat[adjmat_,adjmatg_,adjmatd_,adjmatdg_]:=ArrayFlatten[{{Table[If[i==j,1,If[adjmat[[i,j]]==1,-1,If[i>j,(Subscript[b, j,i]),(Subscript[b, i,j])]]],{i,1,VertexCount[adjmatg]},{j,1,VertexCount[adjmatg]}],Table[If[MemberQ[IGFaces[adjmatg][[j]],i],0,(Subscript[b, j+VertexCount[adjmatg],i])],{i,VertexCount[adjmatg]},{j,VertexCount[adjmatdg]}]},{Transpose[Table[If[MemberQ[IGFaces[adjmatg][[j]],i],0,(Subscript[b, j+VertexCount[adjmatg],i])],{i,VertexCount[adjmatg]},{j,VertexCount[adjmatdg]}]],Table[If[i==j,1,If[adjmatd[[i,j]]==1,-1,If[i>j,(Subscript[b, j+VertexCount[adjmatg],i+VertexCount[adjmatg]]),(Subscript[b, i+VertexCount[adjmatg],j+VertexCount[adjmatg]])]]],{i,1,VertexCount[adjmatdg]},{j,1,VertexCount[adjmatdg]}]}}]

testshape=Normal[AdjacencyMatrix[{1\[UndirectedEdge]2,1\[UndirectedEdge]3,1\[UndirectedEdge]4,1\[UndirectedEdge]6,2\[UndirectedEdge]3,2\[UndirectedEdge]4,2\[UndirectedEdge]5,3\[UndirectedEdge]5,3\[UndirectedEdge]6,4\[UndirectedEdge]5,4\[UndirectedEdge]6,5\[UndirectedEdge]6}]]

pgraph[testshape]
dgraph[testshape]
dadjmat[testshape]

exocta=exGmat[testshape,pgraph[testshape],dadjmat[testshape],dgraph[testshape]]

Solve[Minors[exocta,5]==0,DeleteDuplicates[Cases[exocta,(Subscript[b, _,_]),2]]]

DeleteDuplicates[Cases[exocta,(Subscript[b, _,_]),2]]