We define NLC to be the restriction of the class of graphs NLC k , where relabelling functions are exclusively taken from a set of functions from {1,...,k} into {1,...,k}. We characterize the sets of functions for which NLC is well-quasi-ordered by the induced subgraph relation ≤ i . Precisely, these sets are those which satisfy that for every , we have Im(f ∘ g) = Im(f) or Im(g ∘ f) = Im(g). To obtain this, we show that words (or trees) on are well-quasi-ordered by a relation slightly more constrained than the usual subword (or subtree) relation. A class of graphs is n-well-quasi-ordered if the collection of its vertex-labellings into n colors forms a well-quasi-order under ≤ i , where ≤ i respects labels. Pouzet (C R Acad Sci, Paris Sér A–B 274:1677–1680, 1972) conjectured that a 2-well-quasi-ordered class closed under induced subgraph is in fact n-well-quasi-ordered for every n. A possible approach would be to characterize the 2-well-quasi-ordered classes of graphs. In this respect, we conjecture that such a class is always included in some well-quasi-ordered NLC for some family . This would imply Pouzet’s conjecture.