  Future quantum networks can facilitate communication of quantum information
between various nodes. We are particularly interested in whether multiple pairs
can communicate simultaneously across a network. Quantum networks can be
represented with graph states, and producing communication links amounts to
performing certain quantum operations on graph states. This problem can be
formulated in a graph-theoretic sense with the (Bell) vertex-minor problem. We
discuss the recently introduced foliage partition and provide a generalization.
This generalization leads us to a useful result for approaching the
vertex-minor problem. We apply this result to identify the exact solution for
the Bell vertex-minor problem on line, tree, and ring graphs.
