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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 Sx, 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.

S1={ A, C, D, }
S2={ B, C, D, }
S3={ C, E, }
S4={ A, D, E, }
S5={ B, C, }
S6={ 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