  In this work, we present a methodology for devising forward-backward methods
for finding zeros in the sum of a finite number of maximally monotone
operators. We extend the framework and techniques from [SIAM J. Optim., 34
(2024), pp. 1569-1594] to cover the case involving a finite number of
cocoercive operators, which should be directly evaluated instead of computing
their resolvent. The algorithms are induced by three graphs that determine how
the algorithm variables interact with each other and how they are combined to
compute each resolvent. The hypotheses on these graphs ensure that the
algorithms obtained have minimal lifting and are frugal, meaning that the
ambient space of the underlying fixed point operator has minimal dimension and
that each resolvent and each cocoercive operator is evaluated only once per
iteration. This framework not only allows to recover some known methods, but
also to generate new ones, as the forward-backward algorithm induced by a
complete graph. We conclude with a numerical experiment showing how the choice
of graphs influences the performance of the algorithms.
