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Total variation flow, total variation regularization and the taut stringalgorithm are known to be equivalent filters for one-dimensional discretesignals. In addition, the filtered signal simultaneously minimizes a largenumber of convex functionals in a certain neighbourhood of the data. In thisarticle we study the question to what extent this situation remains true in amore general setting, namely for data given on the vertices of a finiteoriented 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 weprove that the minimizer to the corresponding Rudin-Osher-Fatemi (ROF) model onthe graph has the same universal minimality property as in the one-dimensionalsetting. Interestingly, this property is lost, if $J$ is replaced by thediscrete isotropic total variation. Next, we relate the ROF minimizer to thesolution of the gradient flow for $J$. It turns out that, in contrast to theone-dimensional setting, these two problems are not equivalent in general, butconditions for equivalence are available.

Invariant $\varphi$-minimal sets and total variation denoising on graphs