  We investigate the roles of two classes of quenching mechanisms for central
and satellite galaxies in the SDSS ($z<0.075$): those involving the halo and
those involving the formation of a compact centre. For central galaxies with
inner compactness $\Sigma_{\rm 1kpc} \sim 10^{9-9.4}M_{\odot} {\rm kpc}^{-2}$,
the quenched fraction $f_{q}$ is strongly correlated with $\Sigma_{\rm 1kpc}$
with only weak halo mass $M_{\rm h}$ dependence. However, at higher and lower
$\Sigma_{\rm 1kpc}$, sSFR is a strong function of $M_{\rm h}$ and mostly
independent of $\Sigma_{\rm 1kpc}$. In other words, $\Sigma_{\rm 1kpc} \sim
10^{9-9.4} M_{\odot} {\rm kpc}^{-2}$ divides galaxies into those with high sSFR
below and low sSFR above this range. In both the upper and lower regimes,
increasing $M_{\rm h}$ shifts the entire sSFR distribtuion to lower sSFR
without a qualitative change in shape. This is true even at fixed $M_{*}$, but
varying $M_{*}$ at fixed $M_{\rm h}$ adds no quenching information. Most of the
quenched centrals with $M_{\rm h} > 10^{11.8}M_{\odot}$ are dense ($\Sigma_{\rm
1kpc} > 10^{9}~ M_{\odot} {\rm kpc}^{-2}$), suggesting compaction-related
quenching maintained by halo-related quenching. However, 21% are diffuse,
indicating only halo quenching. For satellite galaxies in the outskirts of
halos, quenching is a strong function of compactness and a weak function of
host $M_{\rm h}$. In the inner halo, $M_{\rm h}$ dominates quenching, with
$\sim 90\%$ of the satellites being quenched once $M_{\rm h} >
10^{13}M_{\odot}$. This regional effect is greatest for the least massive
satellites. As demonstrated via semi-analytic modelling with simple
prescriptions for quenching, the observed correlations can be explained if
quenching due to central compactness is rapid while quenching due to halo mass
is slow.
