  Modular graph forms are a class of modular covariant functions which appear
in the genus-one contribution to the low-energy expansion of closed string
scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the
simplifying property that they may be reduced to sums of products of modular
graph forms of strictly lower loop order. In the particular case of dihedral
modular graph forms, a closed form expression for this holomorphic subgraph
reduction was obtained previously by D'Hoker and Green. In the current work, we
extend these results to trihedral modular graph forms. Doing so involves the
identification of a modular covariant regularization scheme for certain
conditionally convergent sums over discrete momenta, with some elements of the
sum being excluded. The appropriate regularization scheme is identified for any
number of exclusions, which in principle allows one to perform holomorphic
subgraph reduction of higher-point modular graph forms with arbitrary
holomorphic subgraphs.
