  This paper investigates the relations between modular graph forms, which are
generalizations of the modular graph functions that were introduced in earlier
papers motivated by the structure of the low energy expansion of genus-one Type
II superstring amplitudes. These modular graph forms are multiple sums
associated with decorated Feynman graphs on the world-sheet torus. The action
of standard differential operators on these modular graph forms admits an
algebraic representation on the decorations. First order differential operators
are used to map general non-holomorphic modular graph functions to holomorphic
modular forms. This map is used to provide proofs of the identities between
modular graph functions for weight less than six conjectured in earlier work,
by mapping these identities to relations between holomorphic modular forms
which are proven by holomorphic methods. The map is further used to exhibit the
structure of identities at arbitrary weight.
