Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs

Abstract: H. Masur and J. Smillie 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 ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible outer automorphism in 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 outer automorphism in Out(F_r)? In the course of our proof we also develop machinery used in "Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups" to fully answer the question in the rank-three case.