  The string corrections of tree-level open-string amplitudes can be described
by Selberg integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. This
allows for a recursion of the $\alpha'$-expansion of tree-level string
corrections in the number of external states using the Drinfeld associator.
While the feasibility of this recursion is well-known, we provide a
mathematical description in terms of twisted de Rham theory and intersection
numbers of twisted forms. In particular, this leads to purely combinatorial
expressions for the matrix representation of the Lie algebra generators
appearing in the KZ equation in terms of directed graphs. This, in turn, admits
efficient algorithms for symbolic and numerical computations using adjacency
matrices of directed graphs and is a crucial step towards analogous recursions
and algorithms at higher genera.
