Submatrix of small rank
Essentially, you are asking for a graph "separator" of a certain kind, and while this doesn't answer your question directly, your question is likely related to the notion of the rankwidth of a graph. You can find more information in this paper, this paper or especially this paper

Note that the notion of rankwidth uses "rank over the finite field $\mathbb{F}_2$", which may or may not be the field you were asking about.