Standing waves in the focusing nonlinear Schrodinger (NLS) equation are considered on a dumbbell graph (two rings attached to a central line segment subject to the Kirchhoff boundary conditions at the junctions). In the small-norm limit, the ground state (the orbitally stable standing wave of the smallest energy at a fixed $L^2$ norm) is represented by a constant solution. However, when the norm is increased, this constant solution undertakes two bifurcations, where the first is the pitchfork (symmetry breaking) bifurcation and the second one is the symmetry preserving bifurcation. As a result of the first symmetry breaking bifurcation, the standing wave becomes more localized in one of the two rings. As a result of the second symmetry preserving bifurcation, the standing wave becomes localized in the central line segment. In the limit of large norm solutions, both standing waves are represented by a truncated solitary wave localized in either the ring or the central line segment. Although the asymmetric wave supported in the ring is a ground state near the symmetry breaking bifurcation of the constant solution, it is the symmetric wave supported in the central line segment which becomes the ground state in the large-norm limit. These results will be reviewed in the context of standing waves on other bounded and unbounded graphs.