By applying an algorithm of Stallings regarding separability of elements in a free group, we give an alternative approach to that of Osborne and Zieschang in describing all primitive elements in the free group of rank 2. As a result, we give a proof of a classical result of Nielsen, used by Osborne and Zieschang in their work, that the only automorphisms of that act trivially on the abelianization are those defined by conjugation. Finally, we compute the probability that a Whitehead graph in rank 2 contains a cut vertex. We show that this probability is approximately , where is the number of edges in the graph.
Matthew Clay. John Conant. Nivetha Ramasubramanian. "Whitehead graphs and separability in rank two." Involve 7 (4) 431 - 452, 2014. https://doi.org/10.2140/involve.2014.7.431