  Total variation flow, total variation regularization and the taut string
algorithm are known to be equivalent filters for one-dimensional discrete
signals. In addition, the filtered signal simultaneously minimizes a large
number of convex functionals in a certain neighbourhood of the data. In this
article we study the question to what extent this situation remains true in a
more general setting, namely for data given on the vertices of a finite
oriented graph and the total variation being $J(f) = \sum_{i,j} |f(v_i) -
f(v_j)|$. Relying on recent results on invariant $\varphi$-minimal sets we
prove that the minimizer to the corresponding Rudin-Osher-Fatemi (ROF) model on
the graph has the same universal minimality property as in the one-dimensional
setting. Interestingly, this property is lost, if $J$ is replaced by the
discrete isotropic total variation. Next, we relate the ROF minimizer to the
solution of the gradient flow for $J$. It turns out that, in contrast to the
one-dimensional setting, these two problems are not equivalent in general, but
conditions for equivalence are available.
