  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 \emph{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 behavior of the
cohomology basis graph under passage to elementary subminors, and show that it
is not well-behaved under edge contraction.
