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# complementGraph -- returns the complement of a graph or hypergraph

## Synopsis

• Usage:
g = complementGraph G
h = complementGraph H
• Inputs:
• G, ,
• H, ,
• Outputs:
• g, , the complement of G, whose edges are the set of edges not in G
• h, , the complement of H, whose edge set is found by taking the complement of each edge of H in the vertex set

## Description

The function complementGraph finds the complement of a graph and hypergraph. Note that this function behaves differently depending upon the type of input. When applied to a graph, complementGraph returns the graph whose edge set is the set of edges not in G. When applied to a hypergraph, the edge set is found by taking the complement of each edge of H in the vertex set.

 i1 : R = QQ[a,b,c,d,e]; i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle i3 : complementGraph c5 -- the graph complement of the 5-cycle o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}} "ring" => R "vertices" => {a, b, c, d, e} o3 : Graph i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph o4 = HyperGraph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}} "ring" => R "vertices" => {a, b, c, d, e} o4 : HyperGraph i5 : complementGraph c5hypergraph o5 = HyperGraph{"edges" => {{c, d, e}, {a, d, e}, {a, b, e}, {b, c, d}, {a, b, c}}} "ring" => R "vertices" => {a, b, c, d, e} o5 : HyperGraph

## Caveat

Notice that complementGraph works differently on graphs versus hypergraphs.

## Ways to use complementGraph :

• complementGraph(Graph)
• complementGraph(HyperGraph)

## For the programmer

The object complementGraph is .