There is a new embedding for the Klein Graph.

Start with the Braced Heptagon. It’s a unit-distance graph. If you make unit-distance 7/1, 7/2 and 7/3 star polygons, you get forced into this figure:

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GraphData[{"BracedHeptagon",{42,1}}]

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I’ll use the vertex and edge lists:

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vv=GraphData[{"BracedHeptagon",{42,1}},"VertexCoordinates"];

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ee=List@@@GraphData[{"BracedHeptagon",{42,1}},"EdgeList"];

Find all the extraordinary lines of 3 vertices:

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ResourceFunction["FindExtraordinaryLines"][vv]

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{{1,8,18},{1,10,17},{2,9,19},{2,11,18},{3,10,20},{3,12,19},{4,11,21},{4,13,20},{5,12,15},{5,14,21},{6,8,15},{6,13,16},{7,9,16},{7,14,17}}

Split into two sets:

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sets={{{1,8,18},{2,9,19},{3,10,20},{4,11,21},{5,12,15},{6,13,16},{7,14,17}},{{1,10,17},{2,11,18},{3,12,19},{4,13,20},{5,14,21},{6,8,15},{7,9,16}}};

Let’s take a look at those. The blue lines have 3 points, but a fourth point is close.

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Graphics[{Green,Line[vv[[#]]]&/@sets[[1]],Blue,Line[vv[[#]]]&/@sets[[2]],Red,Point[vv]}]

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Create some extra lines connecting to the three star heptagons:

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extra=Join[{#,22}&/@Range[7],{#,23}&/@Range[8,14],{#,24}&/@Range[15,21]];

Create new graphs using either set of the extraordinary lines:

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newembed1=Graph[UndirectedEdge@@@Sort[Join[ee,Flatten[Subsets[#,{2}]&/@sets[[1]],1],extra]],VertexCoordinates->Join[Thread[Range[21]->vv],Thread[{23,22,24}->.07CirclePoints[3]]]]

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In[]:=

newembed2=Graph[UndirectedEdge@@@Sort[Join[ee,Flatten[Subsets[#,{2}]&/@sets[[2]],1],extra]],VertexCoordinates->Join[Thread[Range[21]->vv],Thread[{23,22,24}->.07CirclePoints[3]]]]

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