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 $2$.