  Self-sustaining autocatalytic chemical networks represent a necessary, though
not sufficient condition for the emergence of early living systems. These
networks have been formalised and investigated within the framework of RAF
theory, which has led to a number of insights and results concerning the
likelihood of such networks forming. In this paper, we extend this analysis by
focussing on how {\em small} autocatalytic networks are likely to be when they
first emerge. First we show that simulations are unlikely to settle this
question, by establishing that the problem of finding a smallest RAF within a
catalytic reaction system is NP-hard. However, irreducible RAFs (irrRAFs) can
be constructed in polynomial time, and we show it is possible to determine in
polynomial time whether a bounded size set of these irrRAFs contain the
smallest RAFs within a system. Moreover, we derive rigorous bounds on the sizes
of small RAFs and use simulations to sample irrRAFs under the binary polymer
model. We then apply mathematical arguments to prove a new result suggested by
those simulations: at the transition catalysis level at which RAFs first form
in this model, small RAFs are unlikely to be present. We also investigate
further the relationship between RAFs and another formal approach to
self-sustaining and closed chemical networks, namely chemical organisation
theory (COT).
