Subgraph Complementation and Minimum Rank

Abstract

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of $G$ over $\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show that $c_2 (G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise, $\operatorname{mr}(G,\mathbb{F}_2)\leqslant c_2 (G)\leqslant \operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we show that the following are equivalent for any graph $G$ with at least one edge: i. $c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. the adjacency matrix of $G$ is the unique matrix of rank $\operatorname{mr}(G,\mathbb{F}_2)$ which fits $G$ over $\mathbb{F}_2$; iii. there is a minimum collection $\mathscr{C}$ as described in which every vertex appears an even number of times; and iv. for every component $G'$ of $G$, $c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) + 1$. We also show that, for these graphs, $\operatorname{mr}(G,\mathbb{F}_2)$ is twice the minimum number of tricliques whose symmetric difference of edge sets is $E$. Additionally, we provide a set of upper bounds on $c_2(G)$ in terms of the order, size, and vertex cover number of $G$. Finally, we show that the class of graphs with $c_2(G)\leqslant k$ is hereditary and finitely defined. For odd $k$, the sets of minimal forbidden induced subgraphs are the same as those for the property $\operatorname{mr}(G,\mathbb{F}_2)\leq k$, and we exhibit this set for $c_2(G) \leqslant 2$.