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We investigate the virtual specialness of a compact cube complex $X$ thatsplits as a graph of nonpositively curved cube complexes. We prove virtualspecialness of $X$ when each vertex space of $X$ has word-hyperbolic $\pi_1$and $\pi_1X$ has ``finite stature'' relative to its edge groups. The resultsgeneralize the motivating case when tree $\times$ tree lattices are virtualproducts.

Virtual Specialness of certain graphs of special cube complexes