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On a problem of E.  Prisner concerning the biclique operator. (English) Zbl 1003.05048
Summary: The symbol $$K(B,C)$$ denotes a directed graph with vertex set $$B\cup C$$ for two (not necessarily disjoint) vertex sets $$B,C$$ in which an arc goes from each vertex of $$B$$ to each vertex of $$C$$. A subdigraph of a digraph $$D$$ which has this form is called a bisimplex in $$D$$. A biclique in $$D$$ is a bisimplex in $$D$$ which is not a proper subgraph of any other one and in which $$B\not =\emptyset$$ and $$C\not = \emptyset$$. The biclique digraph $$\vec C(D)$$ of $$D$$ is the digraph whose vertex set is the set of all bicliques in $$D$$ and in which there is an arc from $$K(B_1, C_1)$$ to $$K(B_2,C_2)$$ if and only if $$C_1 \cap B_2 \not = \emptyset$$. The operator which assigns $$\vec C(D)$$ to $$D$$ is the biclique operator $$\vec C$$. The paper solves a problem of E. Prisner concerning the periodicity of $$\vec C$$.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
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