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MATHEMATICA BOHEMICA, Vol. 127, No. 3, pp. 371-373 (2002)
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# On a problem of E. Prisner concerning

the biclique operator

## Bohdan Zelinka

* Bohdan Zelinka*, Department of Applied Mathematics, Faculty of Education, Technical University of Liberec, Voronezska 13, 460 01 Liberec, Czech Republic, e-mail: ` bohdan.zelinka@vslib.cz`

**Abstract:**
The symbol $K(B,C)$ denotes a directed graph with the vertex set $B\cup C$ for two (not necessarily disjoint) vertex sets $B,C$ in which an arc goes from each vertex of $B$ into 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 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 $K(B_1, C_1)$ into $K(B_2,C_2)$ 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 $\vec C$.

**Keywords:** digraph, bisimplex, biclique, biclique digraph, biclique operator, periodicity of an operator

**Classification (MSC2000):** 05C20

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