Abstract: We consider the subcritical nonlinear Schrödinger equation on quantum graphs with an attractive potential supported in the compact core, and investigate the existence and the nonexistence of minimizers of the energy at fixed $L^2$-norm, or mass. First we present a theorem that extends existing techniques to determine the existence of ground states in our context. Then we finally reach the following picture: for small and large mass there are Ground States. Moreover, according to the metric features of the graph and to the strength of the potential, we prove that there may be a region of intermediate masses for which there are no Ground States. The study was originally inspired by the research on quantum waveguides, in which the curvature of a thin tube induces an effective attractive potential.