Equivariant cohomology epimorphisms and face ring quotients for Hamiltonian and complexity one GKM${}_4$ manifolds

Given a GKM${}_3$ action of a torus $K$ on a manifold $M$ with GKM graph $\mathrm{\Gamma }$, we show that for any extension of $\mathrm{\Gamma }$ to an abstract GKM graph the corresponding restriction map in equivariant graph cohomology is surjective. While the corresponding statement for extensions of actions is well-known, we observe that this graph-theoretical statement is false in the GKM${}_2$ setting. As a corollary, we obtain a description of the equivariant cohomology ring of Hamiltonian and complexity one GKM${}_4$ actions in terms of generators and relations.