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# getCliques -- returns cliques in a graph

## Synopsis

• Usage:
C = getCliques(G,d)
C = getCliques G
• Inputs:
• G, ,
• d, an integer, representing the size of the cliques desired
• Outputs:
• C, a list, of cliques of size d or, if no d is entered, all cliques

## Description

A clique of a graph is a subset of its vertices which induces a complete subgraph. That is, a set of vertices is a clique if every pair of vertices in the set forms an edge of the graph. This function returns all cliques of a specified size, and if no size is given, it returns all cliques. Note that all the edges of the graph are considered cliques of size two.

 i1 : R = QQ[a..d]; i2 : G = completeGraph R o2 = Graph{"edges" => {{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}}} "ring" => R "vertices" => {a, b, c, d} o2 : Graph i3 : getCliques(G,3) o3 = {{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}} o3 : List i4 : getCliques(G,4) o4 = {{a, b, c, d}} o4 : List i5 : getCliques G o5 = {{a, b}, {a, c}, {b, c}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, ------------------------------------------------------------------------ {a, c, d}, {b, c, d}, {a, b, c, d}} o5 : List