complementGraph -- returns the complement of a graph or hypergraph

Synopsis

Usage:

g = complementGraph G

h = complementGraph H
Inputs:
- G, a graph,
- H, a hypergraph,
Outputs:
- g, a graph, the complement of G, whose edges are the set of edges not in G
- h, a hypergraph, 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, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
           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