The main theorem of this document emulates, in the context of Out(F_r)
theory, a mapping class group theorem (by H. Masur and J. Smillie) that
determines precisely which index lists arise from pseudo-Anosov mapping
classes. Since the ideal Whitehead graph gives a finer invariant in the
analogous setting of a fully irreducible outer automorphism, we instead focus
on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal
Whitehead graphs of ageometric, fully irreducible outer automorphisms of the
free group of rank 3. Our main theorem accomplishes this by showing that there
are precisely 18 graphs arising as such. We also give a method for identifying
certain complications called periodic Nielsen paths, prove the existence of
conveniently decomposed representatives of ageometric, fully irreducible outer
automorphisms having connected, loop-free, (2r-1)-vertex ideal Whitehead
graphs, and prove a criterion for identifying representatives of ageometric,
fully irreducible outer automorphisms. The methods we use for constructing
fully irreducible outer automorphisms of free groups, as well as our
identification and decomposition techniques, can be used to extend our main
theorem, as they are valid in any rank. Our methods of proof rely primarily on
Bestvina-Feighn-Handel train track theory and the theory of attracting
laminations
