Find a subgraph of k vertices where all vertices are connected to all other vertices?

Question

Given a graph g I would like to find a subgraph sg of k vertices in g such that each vertex in sg is connected to all other vertices in sg. Can this be done in Mathematica?

So far I found the function KCoreComponents, but it seems to be doing something slightly different from what I want.

As an example, let's say k=6 and we can take the sample graph g:

g = AdjacencyGraph[{{0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1,
   1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 
  1, 1, 1, 1, 1}, {0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 
  0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0,
   1, 0, 1, 1, 0, 1, 1, 1, 1, 1}, {0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1,
   0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 
  0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 0, 0, 1, 
  0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0,
   0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0}, {1,
   1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 
  0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1,
   0, 1, 0, 0}, {0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0,
   1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 
  1, 0, 1, 1, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 
  1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0,
   1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 1, 0,
   0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 
  0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0}, {1, 
  1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0,
   1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 
  0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 
  0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0,
   1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 
  1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0}, {0, 0, 1, 1, 1, 1, 1, 
  1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1,
   1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0,
   0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1,
   1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1,
   1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 1, 1, 0, 1, 1, 0, 0,
   1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 
  1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1}, {0, 0, 
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
   1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 
  1, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 
  1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 1, 0, 1, 0, 0}, {1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1,
   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 
  0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0}, {1, 1, 1, 0, 1, 1, 0, 0, 
  0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1,
   0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0}, {0, 1, 1,
   0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 
  1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0,
   1, 1}, {1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1,
   1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 
  0, 1, 0, 1, 0, 1, 0, 0}, {1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0,
   1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1}, {1, 1, 1, 0, 0, 1, 0, 0, 0,
   0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 
  1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0}, {1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1}, {1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 
  1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,
   1, 0, 0, 1, 0, 1, 1}, {1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1,
   0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 
  1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 1, 1, 0, 1, 0, 1, 
  0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0,
   0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1}, {0, 0, 0, 0,
   0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 
  0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0,
   0}, {0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 
  1, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 0, 1, 0,
   1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 
  1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1}, {0, 0, 0, 1, 
  1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1,
   0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 
  1}, {0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1,
   0, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0,
   1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 1, 1, 0, 0,
   1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 
  0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 
  1}, {0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1,
   1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 
  1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0,
   0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1,
   1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 
  0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 
  1}, {0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0,
   0, 1, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1,
   1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 
  0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 
  0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0,
   1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1}, {0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 
  0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 
  1}, {1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 
  1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,
   0, 0, 1, 0, 1, 1}, {1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0,
   0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 
  0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0}, {1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 
  1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1}, {1, 1, 1, 0, 0,
   1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 
  1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 
  0}, {1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 
  0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1,
   1, 1, 0, 1, 0, 0}}];

score 2 · Accepted Answer · answered Jan 04 '22 at 18:25

To find just one such subgraph:

FindClique[g, {6}]
(*    {{14, 15, 19, 25, 33, 41}}    *)

To find lots of them (here, at most 1000):

FindClique[g, {6}, 1000]
(*    {{21, 25, 26, 31, 33, 50}, {21, 25, 26, 31, 33, 49},
       {21, 25, 26, 31, 33, 47}, {21, 25, 26, 31, 33, 43},
       {21, 22, 25, 26, 31, 33}, {9, 21, 25, 26, 31, 33},
       {5, 21, 25, 26, 31, 33}, {13, 21, 25, 26, 31, 49},
       {25, 26, 31, 33, 41, 50}, {25, 26, 31, 33, 41, 49},
       {25, 26, 31, 33, 41, 47}, {25, 26, 31, 33, 41, 43},
       {25, 26, 27, 31, 33, 41}, {24, 25, 26, 31, 33, 41},
       {22, 25, 26, 31, 33, 41}, {19, 25, 26, 31, 33, 41},
       {9, 25, 26, 31, 33, 41}, {7, 25, 26, 31, 33, 41},
       {5, 25, 26, 31, 33, 41}, {4, 25, 26, 31, 33, 41},
       {13, 25, 26, 31, 41, 49}, {1, 25, 26, 31, 41, 49},
       {8, 12, 25, 31, 40, 49}, {3, 8, 12, 25, 31, 49},
       {8, 12, 25, 31, 42, 43}, {8, 12, 25, 31, 40, 43},
       {8, 12, 22, 25, 31, 42}, {8, 12, 22, 25, 31, 40},
       {14, 19, 25, 31, 33, 48}, {14, 16, 25, 31, 33, 50},
       {14, 16, 25, 31, 33, 49}, {14, 16, 25, 31, 33, 43},
       {14, 16, 22, 25, 31, 33}, {9, 14, 16, 25, 31, 33},
       {5, 14, 16, 25, 31, 33}, {14, 19, 25, 31, 33, 41},
       {15, 21, 25, 26, 33, 50}, {15, 21, 25, 26, 33, 49},
       {15, 21, 25, 26, 33, 47}, {15, 21, 25, 26, 33, 43},
       {15, 21, 22, 25, 26, 33}, {9, 15, 21, 25, 26, 33},
       {5, 15, 21, 25, 26, 33}, {13, 15, 21, 25, 26, 49},
       {15, 25, 26, 33, 41, 50}, {15, 25, 26, 33, 41, 49},
       {15, 25, 26, 33, 41, 47}, {15, 25, 26, 33, 41, 43},
       {15, 25, 26, 27, 33, 41}, {15, 24, 25, 26, 33, 41},
       {15, 22, 25, 26, 33, 41}, {15, 19, 25, 26, 33, 41},
       {9, 15, 25, 26, 33, 41}, {7, 15, 25, 26, 33, 41},
       {5, 15, 25, 26, 33, 41}, {4, 15, 25, 26, 33, 41},
       {13, 15, 25, 26, 41, 49}, {1, 15, 25, 26, 41, 49},
       {8, 12, 15, 25, 40, 49}, {3, 8, 12, 15, 25, 49},
       {8, 12, 15, 25, 42, 43}, {8, 12, 15, 25, 40, 43},
       {8, 12, 15, 22, 25, 42}, {8, 12, 15, 22, 25, 40},
       {14, 15, 19, 25, 33, 48}, {14, 15, 16, 25, 33, 50},
       {14, 15, 16, 25, 33, 49}, {14, 15, 16, 25, 33, 43},
       {14, 15, 16, 22, 25, 33}, {9, 14, 15, 16, 25, 33},
       {5, 14, 15, 16, 25, 33}, {14, 15, 19, 25, 33, 41}}    *)

Find a subgraph of k vertices where all vertices are connected to all other vertices?

1 Answers1