  The ellipticity graph of a free group $F$ was defined by I. Kapovich and M.
Lustig in order to study the outer automorphism group of $F$, which acts on
this graph. The graph was constructed to be analogous to the curve complex of a
surface. It is a bipartite graph, whose vertices are conjugacy classes of
nontrivial elements of $F$ and equivalence classes of proper free product
decompositions of the form $F=A*B$. A conjugacy class is joined by an edge to a
free product decomposition $A*B$ whenever the conjugacy class has a
representative in either $A$ or $B$. This paper uses Stallings subgroup
$X$-digraphs and Whitehead automorphisms to construct algorithms that determine
when the distance between two vertices of the ellipticity graph is two, for
both types of vertices.
