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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 allbi-infinite walks through $D$ and $\sigma_D$ is the shift operator. Twodigraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ suchthat $(X_{D_1}, \sigma_{D_1}^n)$ and $(X_{D_2}, \sigma_{D_2}^n)$ aretopologically conjugate for all $n\geq m$. They are called strong shiftequivalent if this holds for $m=1$.  In this paper we introduce a new category which generalises the category ofsubshifts of finite type and topological conjugacy. Our category gives two newnotions of equivalence for digraphs which we call UDAF equivalence and weakUDAF equivalence. UDAF equivalence is a coarser analogue of strong shiftequivalence and weak UDAF equivalence is a coarser analogue of shiftequivalence. Both UDAF and weak UDAF equivalence still separate the n-leafroses for all $n\geq 2$ (the 1-vertex n-edge digraphs). However, UDAFequivalence does not imply shift equivalence, in particular it equates the2-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 groupsof specific objects in our category.  We explain a few equivalent ways of viewing UDAF equivalence, and give anexample of how they can be used to show that Ashley's 8 vertex digraph is UDAFequivalent to the 2-leaf rose.

Unindexed subshifts of finite type and their connection to automorphisms of Thompson's groups