  We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers
in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is
quasi-isometric to a tree. This implies that a group $G$ acting properly and
cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane
stabilizers. Using disc diagram techniques and Wright's recent result on the
aymptotic 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. More precisely, we prove
asymptotic finite-dimensionality for finitely-generated groups acting on
finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded
asymptotic dimension. Finally, we apply contact graph techniques to prove a
cubical version of the flat plane theorem stated in terms of complete bipartite
subgraphs of $\Gamma$.
