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%
%  Find all possible cliques in a graph. 
%  A clique is a sub-graph that is 
%  complete, i.e., any two vertices are connected.
%
%  An undirected graph is represented by the set V of its 
%  vertices and the set E of its edges. Each edge is 
%  represented by the set of the two connected vertices.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


clique(V,E,Clique) :-
    subset(Clique,V) &
    forall(I in Clique,forall(J in Clique,{I,J} in {{I} \ E})).


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%
% A sample graph
%
 graph(g1,{{a,b},{a,c},{b,c},{b,d},{c,d}}).
%
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%
% A sample goal
%
/*
{log}=> graph(g1,G) & clique({a,b,c,d},G,Clique).

G = {{a,b},{a,c},{b,c},{b,d},{c,d}},  
Clique = {}
Clique = {d}
Clique = {c}
Clique = {c,d}
Clique = {b}
Clique = {b,d}
Clique = {b,c}
Clique = {b,c,d}
Clique = {a}
Clique = {a,c}
Clique = {a,b}
Clique = {a,b,c}
no
*/