Masur and Smillie, 1993, proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher, 2011, ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible ϕ ∈ Out(F_r)? We answer this question in the negative by exhibiting, for each r, examples of connected (2r-1)-vertex graphs that are not the ideal Whitehead graph of any fully irreducible ϕ ∈ Out(F_r). In the course of our proof we also develop machinery used in Pfaff, 2012, to fully answer the question in the rank-three case.