The command ComplementHypergraph(H) returns the complement hypergraph of H.

$\mathrm{with}\left(\mathrm{Hypergraphs}\right)\:$$\mathrm{with}\left(\mathrm{ExampleHypergraphs}\right)\:$

$\mathrm{F4}≔\mathrm{Fan}\left(4\right)\;$$\mathrm{Vertices}\left(\mathrm{F4}\right)\;$$\mathrm{Hyperedges}\left(\mathrm{F4}\right)$

$\mathrm{F4}≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 5\; hyperedges\; >}$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\right]$

$\left[\left\{1\&comma;5\right\}\&comma;\left\{2\&comma;5\right\}\&comma;\left\{3\&comma;5\right\}\&comma;\left\{4\&comma;5\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;4\right\}\right]$

$\mathrm{Draw}\left(\mathrm{F4}\right)$

$C≔\mathrm{ComplementHypergraph}\left(\mathrm{F4}\right)\&semi;$$\mathrm{Vertices}\left(C\right)\&semi;$$\mathrm{Hyperedges}\left(C\right)$

$C≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 5\; hyperedges\; >}$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\right]$

$\left[\left\{5\right\}\&comma;\left\{1\&comma;2\&comma;3\right\}\&comma;\left\{1\&comma;2\&comma;4\right\}\&comma;\left\{1\&comma;3\&comma;4\right\}\&comma;\left\{2\&comma;3\&comma;4\right\}\right]$

$\mathrm{Draw}\left(C\right)$

$\mathrm{K2}≔\mathrm{Kuratowski}\left(\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\right\}\&comma;2\right)\&semi;$$\mathrm{K3}≔\mathrm{Kuratowski}\left(\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\right\}\&comma;3\right)$

$\mathrm{K2}≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 10\; hyperedges\; >}$

$\mathrm{K3}≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 10\; hyperedges\; >}$

$\mathrm{Draw}\left(\mathrm{K2}\right)$

$\mathrm{Draw}\left(\mathrm{K3}\right)$

$C≔\mathrm{ComplementHypergraph}\left(\mathrm{K2}\right)\&semi;$$\mathrm{AreEqual}\left(C\&comma;\mathrm{K3}\right)$

$C≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 10\; hyperedges\; >}$

$\mathrm{true}$

$C≔\mathrm{ComplementHypergraph}\left(\mathrm{K3}\right)\&semi;$$\mathrm{AreEqual}\left(C\&comma;\mathrm{K2}\right)$

$C≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 10\; hyperedges\; >}$

$\mathrm{true}$

Claude Berge. Hypergraphes. Combinatoires des ensembles finis. 1987,  Paris, Gauthier-Villars, translated to English.

Claude Berge. Hypergraphs. Combinatorics of Finite Sets.  1989, Amsterdam, North-Holland Mathematical Library, Elsevier, translated from French.

Charles Leiserson, Liyun Li, Marc Moreno Maza and Yuzhen Xie " Parallel computation of the minimal elements of a poset." Proceedings of the 4th International Workshop on Parallel Symbolic Computation (PASCO) 2010: 53-62, ACM.

The Hypergraphs[ComplementHypergraph] command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.