Momentum graphs, Chinese remainder theorem and the surjectivity of the restriction map

We consider a Chinese remainder theorem for (labeled) graphs. For $X$ a GKM $T$-variety and $Y$ an invariant subvariety, we use this to give a condition for surjectivity of the restriction map $H^*(X) \to H^*(Y)$. In particular, this applies to certain invariant subvarieties in a smooth toric variety.