  This paper deals with extensions of vector-valued functions on finite graphs
fulfilling distinguished minimality properties. We show that so-called lex and
L-lex minimal extensions are actually the same and call them minimal Lipschitz
extensions. Then we prove that the solution of the graph $p$-Laplacians
converge to these extensions as $p\to \infty$. Furthermore, we examine the
relation between minimal Lipschitz extensions and iterated weighted midrange
filters and address their connection to $\infty$-Laplacians for scalar-valued
functions. A convergence proof for an iterative algorithm proposed by Elmoataz
et al.~(2014) for finding the zero of the $\infty$-Laplacian is given. Finally,
we present applications in image inpainting.
