I am interested in the problem of expressing the edges of a given (undirected, simple) graph as the sum of edge sets of cliques modulo $2$.

To be more concrete, given a graph $G=(V,E)$, I am seeking to find cliques $C_1=(V_1,E_1)$, $\dots$, $C_k=(V_1,E_k)$ so that $V_i\subseteq V$ for all $i$, each edge set $E_i$ consists of all edges between pairs of vertices of $V_i$, and (most importantly), every edge $e\in E$ lies in an odd number of $E_i$ and every non-edge $e\notin E$ lies in an even number of $E_i$.

For example, in this sense we can express the claw as the sum (modulo $2$) of $K_4$ and $K_3$: (And of course any graph $G=(V,E)$ is the sum of $|E|$ copies of $K_2$, but many graphs have less trivial expressions as sums.)

Have you seen this sort of question asked in the literature? I have not been able to find any terminology for this question, or any literature on it, so I would appreciate almost anything MathOverflow users could share with me about it.