  It follows from the GKM description of equivariant cohomology that the GKM
graph of a GKM manifold has free equivariant graph cohomology, and satisfies a
Poincar\'e duality condition. We prove that these conditions are sufficient for
an abstract $3$-valent $T^2$-GKM graph to be realizable by a simply-connected
$6$-dimensional GKM manifold. Our realization has the property that any closed
stratum of a finite isotropy group contains a fixed point. Furthermore, we
argue that in case there exists a fixed point in whose vicinity there occur at
most two distinct finite nontrivial isotropy groups such a realization is
unique up to equivariant homeomorphism, thus establishing a complexity one GKM
correspondence in dimension $6$. We show that the statement on equivariant
uniqueness is false without the two conditions on the finite isotropies by
providing counterexamples in presence of a fixed point with three distinct
neighbouring finite isotropy groups, as well as an example of a
simply-connected integer GKM manifold with a closed stratum of a finite
isotropy group which does not contain any fixed point.
