On graphs representing automorphisms of free groups

Normal Forms for Basis-Conjugating Automorphisms of a Free Group

“…There are several alternative constructions of Gersten graphs in the literature. For our purposes, an easily constructed infinite version, introduced in [25], is the most suitable.…”

A graph-theoretic proof for Whitehead's second free-group algorithm

“…Proof (Krstić(1989), here streamlined). For each finite 1-containing subset W of F , we let XW and YW denote the vertex-sets of the 1-containing components of the forests X W and W Y respectively; also, we let XW and YW denote the vertex-sets of the tree-closures of W in the trees X F and F Y respectively, where the tree-closure of W in a tree is the inclusion-smallest subtree which includes W .…”

“…Gersten(1987) constructed a graph model for each F -automorphism, and Stallings(1983) pointed out a connection between Gersten's model and Whitehead's. Krstić(1989) used Cayley trees to simplify Gersten's construction. Hoare(1990) gave an explicit description of Whitehead's model in terms of Gersten's.…”