  We study the correlation between galaxy structure and the quenching of star
formation using a sample of SDSS central galaxies with stellar masses 9.75< log
M_*/M_sun<11.25 and redshifts z<0.075. GALEX UV data are used to cleanly divide
the sample into star-forming and quenched galaxies, and to identify galaxies in
transition (the green valley). Despite a stark difference in visual appearance
between blue and red galaxies, their average radial stellar mass density
profiles are remarkably similar (especially in the outer regions) at fixed
mass. The inner stellar mass surface density within a radius of 1 kpc,
\Sigma_1, is used to quantify the growth of the bulge as galaxies evolve. When
galaxies are divided into narrow mass bins, their distribution in the
color-\Sigma_1 plane at fixed mass forms plausible evolutionary tracks.
\Sigma_1 seems to grow as galaxies evolve through the blue cloud, and once it
crosses a threshold value, galaxies are seen to quench at fixed \Sigma_1. The
\Sigma_1 threshold for quenching grows with stellar mass, \Sigma_1 ~
M_*^{0.64}. However, the existence of some star-forming galaxies above the
threshold \Sigma_1 implies that a dense bulge is necessary but not sufficient
to quench a galaxy fully. This would be consistent with a two-step quenching
process in which gas within a galaxy is removed or stabilized against star
formation by bulge-driven processes (such as a starburst, AGN feedback, or
morphological quenching), whereas external gas accretion is suppressed by
separate halo-driven processes (such as halo gas shock heating). Quenching thus
depends on an interplay between the inner structure of a galaxy and its
surrounding dark matter halo, and lack of perfect synchrony between the two
could produce the observed scatter in color vs. \Sigma_1. (Abridged)
