Subgraph Complementation and Minimum Rank

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets $V$ such that $uv\in E$ if and only $u$ $v$ appear together in an odd number sets $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality collection. This invariant is equivalent to dimension faithful orthogonal representation $G$ over $\mathbb{F}_2$ closely connected rank $G$. We show $c_2(G) \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ odd, or forest. Otherwise, $\operatorname{mr}(G,\mathbb{F}_2)\leq c_2(G)\leq \operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we following are for any with at least one edge: i. $c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. adjacency matrix unique which fits $\mathbb{F}_2$; iii. there as described every vertex appears even times; iv. component $G'$ $G$, $c_2(G') \operatorname{mr}(G',\mathbb{F}_2) + 1$. also that, these graphs, twice tricliques whose symmetric difference edge $E$. Additionally, provide set upper bounds on terms order, size, cover Finally, class graphs $c_2(G)\leq k$ hereditary finitely defined. For $k$, minimal forbidden induced subgraphs same those property k$, exhibit this $c_2(G)\leq2$.