  Charm baryon decay plays an important role in studying non-perturbative
baryonic transition. Compared to other hadron multiplets, the flavor symmetry
of baryon decuplet is more simple and attractive. In this work, we study the
topological amplitudes of charmed baryon decays into decuplet baryon in the
flavor symmetry and the linear $SU(3)_F$ breaking. It is found most of
topological diagrams are suppressed by the Korner-Pati-Woo theorem in the
$SU(3)_F$ limit. Only two independent amplitudes contributing to the
$\mathcal{B}_{c\overline 3}\to \mathcal{B}_{10}M$ decays and one of them is
dominated in the branching fractions. The Lee-Yang parameters of all
$\mathcal{B}_{c\overline 3}\to \mathcal{B}_{10}M$ modes are the same in the
$SU(3)_F$ limit, and there are only four possible values for the CP
asymmetries. After including the first-order $SU(3)_F$ breaking effects, the
$\Xi^+_c\to \Sigma^{* +}\overline K^0$ and $\Xi^+_c\to \Xi^{* 0}\pi^+$ decays
have non-zero branching fractions. The amount of free parameters contributing
to the $\mathcal{B}_{c\overline 3}\to \mathcal{B}_{10}M$ decays in the linear
$SU(3)_F$ breaking is smaller than the available data. The $SU(3)_F$ breaking
part of quark loop diagram can be extracted by a global fitting of branching
fractions, which could help us to understand the CP violation in charm sector.
Besides, some new isospin equations are proposed to test the Korner-Pati-Woo
theorem and the isospin symmetry.
