Abstract: The problem discussed in this talk comes from efforts to understand approximation of quantum graph Hamiltonians by Laplacians on "fat graphs". After reviewing the background and known result in both the Neumann and Dirichlet setting we discuss how quantum graphs with nontrivial spectral properties can be obtained from squeezed Dirichlet networks. To illustrate the propose strategy we work out the simplest nontrivial example, a family of bent tubes giving a graph of one vertex and two edges, or a two-parameter family of generalized point interactions on the line.