Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).

Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.

**Case $1$**: $M+W\in\{0,1\}^{n\times n}$.

Could there be tight upper, lower bounds on $rank(M+W)-rank(M)$ depending on only $r$?

**Case $2$**: $M-W\in\{0,1\}^{n\times n}$.

Could there be tight upper, lower bounds on $rank(M-W)-rank(M)$ depending on only $r$?

Above can be reduced to rank $2$ matrices $W$ with symmetric cases.

Matrix $$0\mbox{ }M'$$$$M\mbox{ }0$$ is symmetric and has rank $2r$.

Matrix $$0\mbox{ }W'$$$$W\mbox{ }0$$ is symmetric and has rank $2$.