Delete the subsets containing the same $2$ integers present in other subsets

Question

From my previous question, if I consider a list like this:

$\{$$\{$$\{$$1,2,3$$\}$,$\{$$4,5,6$$\}$$\}$, $\{$$\{$$1,2,4$$\}$,$\{$$3,5,6$$\}$$\}$, $\{$$\{$$1,2,5$$\}$,$\{$$3,4,6$$\}$$\}$, $\{$$\{$$1,2,6$$\}$,$\{$$3,4,5$$\}$$\}$, $\{$$\{$$1,3,4$$\}$,$\{$$2,5,6$$\}$$\}$, $\{$$\{$$1,3,5$$\}$,$\{$$2,4,6$$\}$$\}$, $\{$$\{$$1,3,6$$\}$,$\{$$2,4,5$$\}$$\}$, $\{$$\{$$1,4,5$$\}$,$\{$$2,3,6$$\}$$\}$, $\{$$\{$$1,4,6$$\}$,$\{$$2,3,5$$\}$$\}$, $\{$$\{$$1,5,6$$\}$,$\{$$2,3,4$$\}$$\}$$\}$

how can I delete all the permuted sublists containing in one of their subset $2$ different integers already present in one of the subsets of the previous permuted sublists? I hope the request is clear. In the showed case, the output would just be:

$\{$$\{$$1,2,3$$\}$,$\{$$4,5,6$$\}$$\}$

While considering the sublists of $6$ elements divided in subsets of length $2$, starting with

$\{$$\{$$1,2$$\}$,$\{$$3,4$$\}$,$\{$$5,6$$\}$$\}$

this one has to be deleted:

$\{$$\{$$1,3$$\}$,$\{$$2,4$$\}$,$\{$$5,6$$\}$$\}$

while this one (and others too) should be in the output:

$\{$$\{$$1,3$$\}$,$\{$$4,5$$\}$,$\{$$2,6$$\}$$\}$

Answer 1

This works for the example you gave:

d = {{{1, 2, 3}, {4, 5, 6}}, {{1, 2, 4}, {3, 5, 6}}, {{1, 2, 5}, {3, 
      4, 6}}, {{1, 2, 6}, {3, 4, 5}}, {{1, 3, 4}, {2, 5, 6}}, {{1, 3, 
      5}, {2, 4, 6}}, {{1, 3, 6}, {2, 4, 5}}, {{1, 4, 5}, {2, 3, 
      6}}, {{1, 4, 6}, {2, 3, 5}}, {{1, 5, 6}, {2, 3, 4}}};;
i = 1;
While[++i <= Length[d],
If[Or @@ 
  Flatten[Outer[Length[Intersection[#1, #2]] > 1 &, 
    Flatten[d[[;; i - 1]], 1], d[[i]], 1]], d = Delete[d, i]; --i;]

  ];
d

{{{1, 2, 3}, {4, 5, 6}}}

or:

d = {{{1, 2}, {3, 4}, {5, 6}}, {{1, 3}, {2, 5}, {4, 6}}, {{1, 4}, {2, 
     3}, {5, 6}}, {{1, 5}, {2, 4}, {3, 6}}, {{1, 6}, {2, 3}, {4, 5}}};

results in:

{{{1, 2}, {3, 4}, {5, 6}}, {{1, 3}, {2, 5}, {4, 6}}, {{1, 5}, {2, 
   4}, {3, 6}}, {{1, 6}, {2, 3}, {4, 5}}}

Answer 2

You can use two-argument form of DeleteDuplicates:

ClearAll[cleanUp]

cleanUp = DeleteDuplicates[#, 
    GreaterThan[1] @ Max @ Outer[Length @* Intersection, ##, 1] &] &;

Examples:

d1 = {{{1, 2, 3}, {4, 5, 6}}, {{1, 2, 4}, {3, 5, 6}}, {{1, 2, 5}, {3, 4, 6}},
     {{1, 2, 6}, {3, 4, 5}}, {{1, 3, 4}, {2, 5, 6}}, {{1, 3, 5}, {2, 4, 6}}, 
     {{1, 3, 6}, {2, 4, 5}}, {{1, 4, 5}, {2, 3, 6}}, {{1, 4, 6}, {2, 3, 5}}, 
     {{1, 5, 6}, {2, 3, 4}}};

cleanUp @ d1

{{{1, 2, 3}, {4, 5, 6}}}

d2 = {{{1, 2}, {3, 4}, {5, 6}}, {{1, 3}, {2, 5}, {4, 6}}, {{1, 4}, {2,
      3}, {5, 6}}, {{1, 5}, {2, 4}, {3, 6}}, {{1, 6}, {2, 3}, {4, 5}}};

cleanUp @ d2

{{{1, 2}, {3, 4}, {5, 6}},  
{{1, 3}, {2, 5}, {4, 6}},   
{{1, 5}, {2, 4}, {3, 6}},   
{{1, 6}, {2, 3}, {4, 5}}}