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

J.H.C. Whitehead's second free-group algorithm determines whether or not two given elements of a free group lie in the same orbit of the automorphism group of the free group. The algorithm involves certain connected graphs, and Whitehead used three-manifold models to prove their connectedness; later, Rapaport and Higgins & Lyndon gave group-theoretic proofs. Combined work of Gersten, Stallings, and Hoare showed that the three-manifold models may be viewed as graphs. We give the direct translation of Whitehead's topological argument into the language of graph theory.