  By proving precisely which singularity index lists arise from the pair of
invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and
Smillie determined a Teichm\"uller flow invariant stratification of the space
of quadratic differentials. In this final paper of a three-paper series, we
give a first step to an $Out(F_r)$ analog of the Masur-Smillie theorem. Since
the ideal Whitehead graphs defined by Handel and Mosher give a strictly finer
invariant in the analogous $Out(F_r)$ setting, we determine which of the
twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs
of fully irreducible outer automorphisms in $Out(F_3)$.
