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WolframInstitute`Hypergraph`
HypergraphInsertion  |
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Inserting a hypergraph hg' at a vertex v of a hypergraph hg identifies v with the root of hg', which by convention is the first vertex in [hg']; the rest of hg''s vertices are added as fresh vertices attached the way they were attached to the root in hg'.
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is the raw building block behind : the bracket symmetrizes (or antisymmetrizes) 's results over every ordering of its arguments and collects the outcomes up to isomorphism, while itself returns the unsymmetrized, uncollected list of insertion terms.
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Each returned association pairs a resulting hypergraph with an integer Koszul sign (see ) that keeps track of the sign incurred by inserting into the target's vertex ordering.
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Insertion at a vertex is intended to happen only when that vertex's "Degree" annotation matches the degree of the root of the inserted hypergraph (see ); every vertex defaults to degree 0, so plain hypergraphs (with no degree annotations) insert at every vertex.
Examples
(5)
Basic Examples
(3)
Insert a single edge into every vertex of a triangle:
In[1]:=
Length
[{{1,2,3}}],
[{{1,2}}]
 | HypergraphInsertion |
 | Hypergraph |
| Hypergraph |
Out[1]=
3
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Because a triangle is vertex-transitive, every one of the three insertions gives an isomorphic result:
In[1]:=
EdgeList
First@Keys
[{{1,2,3}}],
[{{1,2}}]1
| CanonicalHypergraph |
 | HypergraphInsertion |
| Hypergraph |
| Hypergraph |
Out[1]=
{{1,4},{2,3,4}}
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Each term carries a Koszul sign; for plain (ungraded) hypergraphs, every sign is 1:
In[1]:=
Values/@
[{{1,2,3}}],
[{{1,2}}]
| HypergraphInsertion |
| Hypergraph |
| Hypergraph |
Out[1]=
{{1},{1},{1}}
Scope
(1)
Possible Issues
(1)
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