  Self-sustaining autocatalytic networks play a central role in living systems,
from metabolism at the origin of life, simple RNA networks, and the modern
cell, to ecology and cognition. A collectively autocatalytic network that can
be sustained from an ambient food set is also referred to more formally as a
`Reflexively Autocatalytic F-generated' (RAF) set. In this paper, we first
investigate a simplified setting for studying RAFs, which are nevertheless
relevant to real biochemistry and allows for a more exact mathematical analysis
based on graph-theoretic concepts. This, in turn, allows for the development of
efficient (polynomial-time) algorithms for questions that are computationally
NP-hard in the general RAF setting. We then show how this simplified setting
for RAF systems leads naturally to a more general notion of RAFs that are
`generative' (they can be built up from simpler RAFs) and for which efficient
algorithms carry over to this more general setting. Finally, we show how
classical RAF theory can be extended to deal with ensembles of catalysts as
well as the assignment of rates to reactions according to which catalysts (or
combinations of catalysts) are available.
