We examine a graph Γ encoding the intersection of hyperplane carriers in a CAT(0) cube complex . The main result is that Γ is quasi-isometric to a tree. This implies that a group G acting properly and cocompactly on is weakly hyperbolic relative to the hyperplane stabilizers. Using Wright's recent result on the asymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of Γ.