Right-angled Artin groups and the cohomology basis graph

Let Gamma be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H<^>1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$ from the cup product, called the cohomology basis graph. We show that $\Gamma_{\mathcal B}$ always contains Gamma as a subgraph. This provides an effective way to reconstruct the defining graph Gamma from the cohomology of $A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of $A(\Gamma)$ and to recover many other natural graph-theoretic invariants. We also investigate the behaviour of the cohomology basis graph under passage to elementary subminors and show that it is not well-behaved under edge contraction.