A characterization of linear operators that preserve isolation numbers

LeRoy B. Beasley, Song Seok-Zun


We obtain  characterizations of Boolean linear operators that preserve some of the isolation numbers  of Boolean matrices.  In particular, we show that the following are equivalent:  (1) $T$ preserves the isolation number of all matrices; (2) $T$  preserves the set of matrices with isolation number one and the set of those with isolation number $k$ for some $2\leq k\leq \min\{m,n\}$; (3) for $1\leq k\leq \min\{m,n\}-1$, $T$ preserves matrices
with isolation number $k$, and those with isolation number $k+1$, (4) $T$ maps $J$ to $J$ and preserves the set of matrices of isolation number 2;  (5)  $T$ is a $(P,Q)$-operator, that is, for fixed permutation matrices $P$ and $Q$,  $m\times n$ matrix $X,$~ $T(X)=PXQ$ or, $m=n$ and  $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.


Boolean matrix, Boolean rank, isolation number, Boolean linear opertator

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ISSN: 1331-0623 (Print), 1848-8013 (Online)