TITLE: Terminology for expressing a graph as a sum of cliques (mod 2)
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: The closest notion to this that I have found in the literature is that of subgraph complementation, introduced by Kamiński, Lozin, and Milanič, in which the edges of a given induced subgraph of a graph $G$ are complemented. Then we may phrase your problem as building $G$ by taking successive subgraph complements, starting with the empty graph on $V(G)$.
In fact, your question inspired a research project amongst myself and the two others who responded to this post, the product of which is available here: arXiv:2101.06180.
We provide upper bounds on the minimum number of cliques in an expression of $G$ as you describe in terms of the number of vertices, the number of edges, and the size of a minimum vertex cover. We relate this problem to the minimum rank problem over the field of order $2$, enabling us in some cases to find the minimum size of such an expression. We also show that, similar to the minimum rank problem over $\mathbb{F}_2$, the class of graphs which may be expressed as a sum of $k$ cliques is hereditary and finitely defined for any positive integer $k$.
By considering an incidence matrix corresponding to a collection of cliques in an expression, we see that your problem is equivalent to that of finding a faithful orthogonal representation of a graph over $\mathbb{F}_2$; that is, an assignment of vectors over $\mathbb{F}_2$ to the vertices of $G$ so that two vectors are orthogonal if and only if they represent non-adjacent vertices. In the study of minimizing the dimension of such a representation, Lozin and Alekseev use an early variant of the subgraph complement (see the proof of Theorem 3).