  Modular graph forms (MGFs) are a class of non-holomorphic modular forms which
naturally appear in the low-energy expansion of closed-string genus-one
amplitudes and have generated considerable interest from pure mathematicians.
MGFs satisfy numerous non-trivial algebraic- and differential relations which
have been studied extensively in the literature and lead to significant
simplifications. In this paper, we systematically combine these relations to
obtain basis decompositions of all two- and three-point MGFs of total modular
weight $w+\bar{w}\leq12$, starting from just two well-known identities for
banana graphs. Furthermore, we study previously known relations in the integral
representation of MGFs, leading to a new understanding of holomorphic subgraph
reduction as Fay identities of Kronecker--Eisenstein series and opening the
door towards decomposing divergent graphs. We provide a computer implementation
for the manipulation of MGFs in the form of the $\texttt{Mathematica}$ package
$\texttt{ModularGraphForms}$ which includes the basis decompositions obtained.
