  We investigate the existence of ground states for the nonlinear Schr\"odinger
Equation on star graphs with two subcritical focusing nonlinear terms: a
standard power nonlinearity, and a delta-type nonlinearity located at the
vertex. We find that if the standard nonlinearity is stronger than the
pointwise one, then ground states exist for small mass only. On the contrary,
if the pointwise nonlinearity prevails, then ground states exist for large mass
only. All ground states are radial, in the sense that their restriction to each
half-line is always the same function, and coincides with a soliton tail.
Finally, if the two nonlinearities are of the same size, then the existence of
ground states is insensitive to the value of the mass, and holds only on graphs
with a small number of half-lines. Furthermore, we establish the orbital
stability of the branch of radial stationary states to which the ground states
belong, also in the mass regimes in which there is no ground state.
