  An updated analysis of the two-body $D\to PP, V\!P$ and $VV$ decays within
the framework of the topological diagram approach is performed. A global fit to
the Cabibbo-favored (CF) modes in the $V\!P$ sector gives many solutions with
similarly small local minima in $\chi^2$. The solution degeneracy is lifted
once we use them to predict for the singly Cabibbo-suppressed (SCS) modes.
Topological amplitudes are extracted for the $\eta-\eta'$ mixing angles
$\phi=40.4^\circ$ and $43.5^\circ$. The $K_S^0-K_L^0$ asymmetries in $D\to
K_{S,L}^0M$ decays denoted by $R(D,M)$ are studied. While the predicted
$R(D^0,P)$ for $P=\pi^0, \eta$ and $\eta'$ agree with experiment, the
calculated $R(D^+,\pi^+)$, $R(D_s^+, K^+)$, $R(D^0,\omega)$ and $R(D^0,\phi)$
deviate from the data. We conjecture that the relative phase between the
topological amplitudes $(C+A)$ and $(T+C)$ should be slightly smaller than
$90^\circ$ in order to explain the first two discrepancies and that additional
singlet contributions due to the SU(3)-singlet nature of $\omega$ and $\phi$
are needed to account for the last two. For doubly Cabibbo-suppressed (DCS)
$D\to V\!P$ decays, their topological amplitudes (double-primed) cannot be all
the same as the corresponding ones in the CF modes. The assumption of
$E_{V,P}''=E_{V,P}$ for the $W$-exchange amplitude leads to some
inconsistencies with the experiment. Through the measured relative phases
between CF and DCS channels, the relations of $E_{V,P}''$ with $E_{V,P}$ are
determined. Long-distance contributions to the $D^0$-$\overline {D}^0$ mixing
parameter $y$ are evaluated in the exclusive approach. In particular, we focus
on $D\to PP$ and $V\!P$ decays where $y$ can be reliably estimated. We conclude
that $y_{_{P\!P}}\sim (0.110\pm 0.011)\%$ and the lower bound on $y_{_{V\!P}}$
is $(0.220\pm 0.071)\%$.
