Algorithmic decidability of the universal equivalence problem for partially commutative nilpotent groups

Let $G_\Gamma$ be a partially commutative group corresponding to a finite simple graph $\Gamma$. Given a finite simple graph $T$, an existential graph formula $\phi(T)$ is constructed. We describe an algorithm that answers the question whether $\phi(T)$ is satisfied on $G_\Gamma$, for an arbitrary simple graph $T$. Using this algorithm, we show that the universal equivalence problem for partially commutative class two nilpotent groups is algorithmically decidable.