On a problem of E. Prisner concerning the biclique operator

Zelinka, Bohdan

Mathematica Bohemica, Tome 127 (2002) no. 3, pp. 371-373.

Résumé

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\ne \emptyset $ and $C\ne \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 \ne \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 $\vec{C}$.

Détail

DOI : 10.21136/MB.2002.134064

Classification : 05C20
Keywords: digraph; bisimplex; biclique; biclique digraph; biclique operator; periodicity of an operator