Examples for the notion of duality
A convenient way to think of duals is by the
incidence matrix of H,
where the columns are labeled by the elements of
A and the rows by the elements of V,
and there is a `1' (or `*`, as in the figures) in row
x and column a if
a S_{x},
otherwise there is a `0`.
Thus the rows `correspond' to the hyperedges of H.
We give a hypergraph by its list of hyperedges, as
Venn diagram, and by its incidence matrix.
S_{1}={ 
A, 

C, 
D, 

} 
S_{2}={ 

B, 
C, 
D, 

} 
S_{3}={ 


C, 

E, 
} 
S_{4}={ 
A, 


D, 
E, 
} 
S_{5}={ 

B, 
C, 


} 
S_{6}={ 
A, 


D, 

} 



A 
B 
C 
D 
E 
1 
* 
0 
* 
* 
0 
2 
0 
* 
* 
* 
0 
3 
0 
0 
* 
0 
* 
4 
* 
0 
0 
* 
* 
5 
0 
* 
* 
0 
0 
6 
* 
0 
0 
* 
0 

By transposing the matrix, the roles of columns and vertices
are interchanged.
The dual H* is the hypergraph whose incidence matrix
equals the transpose of the incidence matrix of H.
Now the dual of the hypergraph above is described as follows:

1 
2 
3 
4 
5 
6 
A 
* 
0 
0 
* 
0 
* 
B 
0 
* 
0 
0 
* 
0 
C 
* 
* 
* 
0 
* 
0 
D 
* 
* 
0 
* 
0 
* 
E 
0 
0 
* 
* 
0 
0 

A*={ 
1, 


4, 

6, 
} 
B*={ 

2, 


5, 

} 
C*={ 
1, 
2, 
3, 

5, 

} 
D*={ 
1, 
2, 

4, 

6, 
} 
E*={ 


3, 
4, 


} 


The
intersection graph of the first hypergraph is
Back to the
start page for intersection graphs.
Erich Prisner
made on January 12, 1999