Automorphisms of graphs of cyclic splittings of free groups
  We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings. 
Camille Horbez and Richard D. Wade
math.GR math.GT