On a problem of E. Prisner concerning the biclique operator
Mathematica Bohemica, Tome 127 (2002) no. 3, pp. 371-373
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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 $\vec{C}$.
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 $\vec{C}$.
DOI :
10.21136/MB.2002.134064
Classification :
05C20
Keywords: digraph; bisimplex; biclique; biclique digraph; biclique operator; periodicity of an operator
Keywords: digraph; bisimplex; biclique; biclique digraph; biclique operator; periodicity of an operator
Zelinka, Bohdan. On a problem of E. Prisner concerning the biclique operator. Mathematica Bohemica, Tome 127 (2002) no. 3, pp. 371-373. doi: 10.21136/MB.2002.134064
@article{10_21136_MB_2002_134064,
author = {Zelinka, Bohdan},
title = {On a problem of {E.~Prisner} concerning the biclique operator},
journal = {Mathematica Bohemica},
pages = {371--373},
year = {2002},
volume = {127},
number = {3},
doi = {10.21136/MB.2002.134064},
mrnumber = {1931321},
zbl = {1003.05048},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.134064/}
}
TY - JOUR AU - Zelinka, Bohdan TI - On a problem of E. Prisner concerning the biclique operator JO - Mathematica Bohemica PY - 2002 SP - 371 EP - 373 VL - 127 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.134064/ DO - 10.21136/MB.2002.134064 LA - en ID - 10_21136_MB_2002_134064 ER -