  Charmed baryon decay plays an important role in studying the weak and strong
interactions. In this work, we investigate the topological diagrams of charmed
baryon anti-triplet decays into light baryon octet and light meson. The
topological diagrams of the $\mathcal{B}_{c\overline 3}\to \mathcal{B}_8^S M$
modes and the diagrams with quark loop are presented for the first time. The
completeness of topologies is confirmed by permutation. The linear correlation
of topologies are obtained by deriving the relation between the topological
amplitudes constructed by the third- and second-rank octet tensors. It is found
the topologies contributing to the $\mathcal{B}_{c\overline 3}\to
\mathcal{B}_8^S M$ modes can be determined by the topologies contributing to
the $\mathcal{B}_{c\overline 3}\to \mathcal{B}_8^A M$ modes, vice versa. The
equations of $SU(3)$ irreducible amplitudes decomposed by topologies are
derived through two different intermediate amplitudes, but the inverse solution
does not exist since the number of topologies is larger than number of $SU(3)$
irreducible amplitudes. Applying this framework to the Standard Model, it is
found there are thirteen independent $SU(3)$ irreducible amplitudes
contributing to the $\mathcal{B}_{c\overline 3}\to \mathcal{B}_8 M$ decays.
Considering the suppressions induced by the small CKM matrix elements and the
Korner-Pati-Woo theorem, the branching fractions of charmed baryon decays are
dominated by five $SU(3)$ irreducible amplitudes in the $SU(3)_F$ limit. The
quark-loop diagrams could enhance the $U$-spin breaking effects and then
increase the branching fraction difference of two decay channels. The
systematic measurements of branching fractions of the singly Cabibbo-suppressed
modes could help us to find which modes are promising in searching for CP
asymmetries in charmed baryon sector.
