  We consider certain elliptic modular graph functions that arise in the
asymptotic expansion around the non--separating node of genus two string
invariants that appear in the integrand of the $D^8 R^4$ interaction in the low
momentum expansion of the four graviton amplitude in type II superstring
theory. These elliptic modular graphs have links given by the Green function,
as well its holomorphic and anti--holomorphic derivatives. Using appropriate
auxiliary graphs at various intermediate stages of the analysis, we show that
each graph can be expressed solely in terms of graphs with links given only by
the Green function and not its derivatives. This results in a reduction in the
number of basis elements in the space of elliptic modular graphs.
