Linear operators that preserve graphical properties of matrices: isolation numbers

@article{Beasley2014,
abstract = {Let $A$ be a Boolean $\lbrace 0,1\rbrace $ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of the set $S$ into the complement of the set $S$. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that $T$ is a Boolean linear operator that strongly preserves isolation number $k$ for any $1\le k\le \min \lbrace m,n\rbrace $ if and only if there are fixed permutation matrices $P$ and $Q$ such that for $X\in \{\mathcal \{M\}\}_\{m,n\}(\mathbb \{B\})$$T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.},
author = {Beasley, LeRoy B., Song, Seok-Zun, Jun, Young Bae},
journal = {Czechoslovak Mathematical Journal},
keywords = {Boolean matrix; Boolean rank; Boolean linear operator; isolation number; Boolean matrix; Boolean rank; Boolean linear operator; isolation number},
language = {eng},
number = {3},
pages = {819-826},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear operators that preserve graphical properties of matrices: isolation numbers},
url = {http://eudml.org/doc/262187},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Beasley, LeRoy B.
AU - Song, Seok-Zun
AU - Jun, Young Bae
TI - Linear operators that preserve graphical properties of matrices: isolation numbers
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 819
EP - 826
AB - Let $A$ be a Boolean $\lbrace 0,1\rbrace $ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of the set $S$ into the complement of the set $S$. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that $T$ is a Boolean linear operator that strongly preserves isolation number $k$ for any $1\le k\le \min \lbrace m,n\rbrace $ if and only if there are fixed permutation matrices $P$ and $Q$ such that for $X\in {\mathcal {M}}_{m,n}(\mathbb {B})$$T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.
LA - eng
KW - Boolean matrix; Boolean rank; Boolean linear operator; isolation number; Boolean matrix; Boolean rank; Boolean linear operator; isolation number
UR - http://eudml.org/doc/262187
ER -