  For a finite digraph $D$, we define the corresponding subshift of finite type
$(X_D, \sigma_D)$ to be the dynamical system where $X_D$ is the set of all
bi-infinite walks through $D$ and $\sigma_D$ is the shift operator. Two
digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such
that $(X_{D_1}, \sigma_{D_1}^n)$ and $(X_{D_2}, \sigma_{D_2}^n)$ are
topologically conjugate for all $n\geq m$. They are called strong shift
equivalent if this holds for $m=1$.
  In this paper we introduce a new category which generalises the category of
subshifts of finite type and topological conjugacy. Our category gives two new
notions of equivalence for digraphs which we call UDAF equivalence and weak
UDAF equivalence. UDAF equivalence is a coarser analogue of strong shift
equivalence and weak UDAF equivalence is a coarser analogue of shift
equivalence. Both UDAF and weak UDAF equivalence still separate the n-leaf
roses for all $n\geq 2$ (the 1-vertex n-edge digraphs). However, UDAF
equivalence does not imply shift equivalence, in particular it equates the
2-leaf rose with the golden mean shift.
  We also show that our category relates to the automorphism groups of the
"V-type" groups of Higman, Thompson and Brin. In particular for all $n\geq 2$
the groups $Out(G_{n, n-1})$ and $Out(nV)$ can be seen as automorphism groups
of specific objects in our category.
  We explain a few equivalent ways of viewing UDAF equivalence, and give an
example of how they can be used to show that Ashley's 8 vertex digraph is UDAF
equivalent to the 2-leaf rose.
