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If we extend this code further, we could build an AdjacencyMatrix in which we want to select a minimum set of nodes that are never incident to the same edge (like FindIndependentVertexSet but in minimum terms):

If we extend this code further, we could build an AdjacencyMatrix in which we want to select a minimum set of nodes that we can reach all the vertices with at most 1 step(like FindIndependentVertexSet but in minimum terms and little refinement, vertices can be two sides of an edge):

graphRules = 
  DeleteDuplicatesBy[Sort]@
   Catenate@
    Table[Thread[
      index \[UndirectedEdge] 
       Catenate@
        Position[pairs, 
         Alternatives @@ 
          Table[ReplacePart[pairs[[index]], i -> _], {i, 
            3}], {1}]], {index, Length@pairs}];

adjacency = Unitize@AdjacencyMatrix[graphRules];

graphRules = 
  DeleteDuplicatesBy[Sort]@
   Catenate@
    Table[Thread[
      index \[UndirectedEdge] 
       Catenate@
        Position[pairs, 
         Alternatives @@ 
          Table[ReplacePart[pairs[[index]], i -> _], {i, 
            3}], {1}]], {index, n}];

adjacency = Unitize@AdjacencyMatrix[graphRules];

Update 2

For checking the correctness you can use:

DeleteCases[pairs, 
 Alternatives @@ (({#1, #2, _} | {#1, _, #3} | {_, #2, #3}) & @@@ chosen)]

Where pairs are all the points and chosen are the picked ones.