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The graph M??OPGacaWICaPwE?

This graph is embeddable. It is even 9-grid embeddable.

Embedding

An embedding of this graph is given by.

  1. (0, 8, 8)
  2. (8, 0, 8)
  3. (4, 4, 8)
  4. (4, 8, 0)
  5. (8, 4, 0)
  6. (4, -2, 8)
  7. (2, -4, -8)
  8. (8, 0, -4)
  9. (0, 8, -4)
  10. (8, -4, 4)
  11. (4, -8, -4)
  12. (8, -4, -8)
  13. (4, -8, 8)
  14. (8, 8, -8)

Reduce script

Embeddability of the graph reduces to the truth of a formula over the reals, which can be decided with the reduce algebra package using the following script.

load_package redlog;
rlset R;

procedure d(x,y);
    (first x) * (first y) +
    (second x) * (second y) +
    (third x) * (third y);

procedure k(x,y);
    {(second x)*(third y) - (third x)*(second y),
     (third x)*(first y) - (first x)*(third y),
     (first x)*(second y) - (second x)*(first y)};

v0c1 := 1; v0c2 := 0; v0c3 := 0;
v1c1 := 0; v1c2 := 1; v1c3 := 0;

v0 := {v0c1, v0c2, v0c3}; 
v1 := {v1c1, v1c2, v1c3}; 
v2 := {v2c1, v2c2, v2c3}; 
neq0 := k(k(k(v2,v1),v1),v2); 
neq1 := k(k(k(v2,v1),v1),k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1))); 
neq2 := k(k(k(v2,v1),v1),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1)); 
neq3 := k(k(k(v2,v1),v1),k(k(k(v2,v1),v2),v0)); 
neq4 := k(k(k(v2,v1),v1),k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2))); 
neq5 := k(k(k(v2,v1),v1),v0); 
neq6 := k(k(k(v2,v1),v1),k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)))); 
neq7 := k(k(k(v2,v1),v1),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0)); 
neq8 := k(k(k(v2,v1),v1),k(k(v2,v1),v2)); 
neq9 := k(k(k(v2,v1),v1),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq10 := k(v2,k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1))); 
neq11 := k(v2,k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1)); 
neq12 := k(v2,k(k(k(v2,v1),v2),v0)); 
neq13 := k(v2,k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2))); 
neq14 := k(v2,v0); 
neq15 := k(v2,k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)))); 
neq16 := k(v2,k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0)); 
neq17 := k(v2,v1); 
neq18 := k(v2,k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq19 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1)); 
neq20 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(v2,v1),v2),v0)); 
neq21 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2))); 
neq22 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),v0); 
neq23 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),v1); 
neq24 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(v2,v1),v2)); 
neq25 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq26 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq27 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(k(v2,v1),v2),v0)); 
neq28 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),v0); 
neq29 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)))); 
neq30 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0)); 
neq31 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)); 
neq32 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq33 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(v2,v1)); 
neq34 := k(k(k(k(v2,v1),v2),v0),k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2))); 
neq35 := k(k(k(k(v2,v1),v2),v0),k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)))); 
neq36 := k(k(k(k(v2,v1),v2),v0),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0)); 
neq37 := k(k(k(k(v2,v1),v2),v0),v1); 
neq38 := k(k(k(k(v2,v1),v2),v0),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq39 := k(k(k(k(v2,v1),v2),v0),k(v2,v1)); 
neq40 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)),v0); 
neq41 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0)); 
neq42 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)),v1); 
neq43 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq44 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq45 := k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2)),k(v2,v1)); 
neq46 := k(v0,k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)))); 
neq47 := k(v0,k(k(v2,v1),v2)); 
neq48 := k(v0,k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq49 := k(v0,k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq50 := k(v0,k(v2,v1)); 
neq51 := k(k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))),k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0)); 
neq52 := k(k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))),v1); 
neq53 := k(k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))),k(k(v2,v1),v2)); 
neq54 := k(k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq55 := k(k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))),k(v2,v1)); 
neq56 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),v1); 
neq57 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(k(v2,v1),v2)); 
neq58 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq59 := k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)); 
neq60 := k(v1,k(k(v2,v1),v2)); 
neq61 := k(v1,k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq62 := k(v1,k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq63 := k(k(k(v2,v1),v2),k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2)); 
neq64 := k(k(k(v2,v1),v2),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))); 
neq65 := k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),k(v2,v1)); 
neq66 := k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),k(v2,v1)); 
phi := 
       (first neq0 neq 0 or
        second neq0 neq 0 or
        third neq0 neq 0) and 
       (first neq1 neq 0 or
        second neq1 neq 0 or
        third neq1 neq 0) and 
       (first neq2 neq 0 or
        second neq2 neq 0 or
        third neq2 neq 0) and 
       (first neq3 neq 0 or
        second neq3 neq 0 or
        third neq3 neq 0) and 
       (first neq4 neq 0 or
        second neq4 neq 0 or
        third neq4 neq 0) and 
       (first neq5 neq 0 or
        second neq5 neq 0 or
        third neq5 neq 0) and 
       (first neq6 neq 0 or
        second neq6 neq 0 or
        third neq6 neq 0) and 
       (first neq7 neq 0 or
        second neq7 neq 0 or
        third neq7 neq 0) and 
       (first neq8 neq 0 or
        second neq8 neq 0 or
        third neq8 neq 0) and 
       (first neq9 neq 0 or
        second neq9 neq 0 or
        third neq9 neq 0) and 
       (first neq10 neq 0 or
        second neq10 neq 0 or
        third neq10 neq 0) and 
       (first neq11 neq 0 or
        second neq11 neq 0 or
        third neq11 neq 0) and 
       (first neq12 neq 0 or
        second neq12 neq 0 or
        third neq12 neq 0) and 
       (first neq13 neq 0 or
        second neq13 neq 0 or
        third neq13 neq 0) and 
       (first neq14 neq 0 or
        second neq14 neq 0 or
        third neq14 neq 0) and 
       (first neq15 neq 0 or
        second neq15 neq 0 or
        third neq15 neq 0) and 
       (first neq16 neq 0 or
        second neq16 neq 0 or
        third neq16 neq 0) and 
       (first neq17 neq 0 or
        second neq17 neq 0 or
        third neq17 neq 0) and 
       (first neq18 neq 0 or
        second neq18 neq 0 or
        third neq18 neq 0) and 
       (first neq19 neq 0 or
        second neq19 neq 0 or
        third neq19 neq 0) and 
       (first neq20 neq 0 or
        second neq20 neq 0 or
        third neq20 neq 0) and 
       (first neq21 neq 0 or
        second neq21 neq 0 or
        third neq21 neq 0) and 
       (first neq22 neq 0 or
        second neq22 neq 0 or
        third neq22 neq 0) and 
       (first neq23 neq 0 or
        second neq23 neq 0 or
        third neq23 neq 0) and 
       (first neq24 neq 0 or
        second neq24 neq 0 or
        third neq24 neq 0) and 
       (first neq25 neq 0 or
        second neq25 neq 0 or
        third neq25 neq 0) and 
       (first neq26 neq 0 or
        second neq26 neq 0 or
        third neq26 neq 0) and 
       (first neq27 neq 0 or
        second neq27 neq 0 or
        third neq27 neq 0) and 
       (first neq28 neq 0 or
        second neq28 neq 0 or
        third neq28 neq 0) and 
       (first neq29 neq 0 or
        second neq29 neq 0 or
        third neq29 neq 0) and 
       (first neq30 neq 0 or
        second neq30 neq 0 or
        third neq30 neq 0) and 
       (first neq31 neq 0 or
        second neq31 neq 0 or
        third neq31 neq 0) and 
       (first neq32 neq 0 or
        second neq32 neq 0 or
        third neq32 neq 0) and 
       (first neq33 neq 0 or
        second neq33 neq 0 or
        third neq33 neq 0) and 
       (first neq34 neq 0 or
        second neq34 neq 0 or
        third neq34 neq 0) and 
       (first neq35 neq 0 or
        second neq35 neq 0 or
        third neq35 neq 0) and 
       (first neq36 neq 0 or
        second neq36 neq 0 or
        third neq36 neq 0) and 
       (first neq37 neq 0 or
        second neq37 neq 0 or
        third neq37 neq 0) and 
       (first neq38 neq 0 or
        second neq38 neq 0 or
        third neq38 neq 0) and 
       (first neq39 neq 0 or
        second neq39 neq 0 or
        third neq39 neq 0) and 
       (first neq40 neq 0 or
        second neq40 neq 0 or
        third neq40 neq 0) and 
       (first neq41 neq 0 or
        second neq41 neq 0 or
        third neq41 neq 0) and 
       (first neq42 neq 0 or
        second neq42 neq 0 or
        third neq42 neq 0) and 
       (first neq43 neq 0 or
        second neq43 neq 0 or
        third neq43 neq 0) and 
       (first neq44 neq 0 or
        second neq44 neq 0 or
        third neq44 neq 0) and 
       (first neq45 neq 0 or
        second neq45 neq 0 or
        third neq45 neq 0) and 
       (first neq46 neq 0 or
        second neq46 neq 0 or
        third neq46 neq 0) and 
       (first neq47 neq 0 or
        second neq47 neq 0 or
        third neq47 neq 0) and 
       (first neq48 neq 0 or
        second neq48 neq 0 or
        third neq48 neq 0) and 
       (first neq49 neq 0 or
        second neq49 neq 0 or
        third neq49 neq 0) and 
       (first neq50 neq 0 or
        second neq50 neq 0 or
        third neq50 neq 0) and 
       (first neq51 neq 0 or
        second neq51 neq 0 or
        third neq51 neq 0) and 
       (first neq52 neq 0 or
        second neq52 neq 0 or
        third neq52 neq 0) and 
       (first neq53 neq 0 or
        second neq53 neq 0 or
        third neq53 neq 0) and 
       (first neq54 neq 0 or
        second neq54 neq 0 or
        third neq54 neq 0) and 
       (first neq55 neq 0 or
        second neq55 neq 0 or
        third neq55 neq 0) and 
       (first neq56 neq 0 or
        second neq56 neq 0 or
        third neq56 neq 0) and 
       (first neq57 neq 0 or
        second neq57 neq 0 or
        third neq57 neq 0) and 
       (first neq58 neq 0 or
        second neq58 neq 0 or
        third neq58 neq 0) and 
       (first neq59 neq 0 or
        second neq59 neq 0 or
        third neq59 neq 0) and 
       (first neq60 neq 0 or
        second neq60 neq 0 or
        third neq60 neq 0) and 
       (first neq61 neq 0 or
        second neq61 neq 0 or
        third neq61 neq 0) and 
       (first neq62 neq 0 or
        second neq62 neq 0 or
        third neq62 neq 0) and 
       (first neq63 neq 0 or
        second neq63 neq 0 or
        third neq63 neq 0) and 
       (first neq64 neq 0 or
        second neq64 neq 0 or
        third neq64 neq 0) and 
       (first neq65 neq 0 or
        second neq65 neq 0 or
        third neq65 neq 0) and 
       (first neq66 neq 0 or
        second neq66 neq 0 or
        third neq66 neq 0) and 
       d(k(k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v0),k(v2,v1)),k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1))),k(k(k(k(k(k(k(v2,v1),v2),v0),k(k(v2,v1),v1)),v2),v1),k(k(v2,v1),v2))) = 0 and 
        true;
rlqe 
     ex(v2c1,
     ex(v2c3,
     ex(v2c2,phi)));