Vertex isoperimetric number of a graph - NP-hard?
The following paper:
On an isoperimetric problem for Hamming graphs
L.H. Harper. contains the following:


  The vertex-isoperimetric problem is NP-complete [8] in
  general, so no polynomially bounded solution is known and it is unlikely that one
  exists.


Where reference [8] is
M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, 

This book usually either contains a proof or a reference to the proof.
Unfortunately I have no access to the book at the moment.