  We are interested in characterizing which classes of finite graphs are
well-quasi-ordered by the induced subgraph relation. To that end, we devise an
algorithm to decide whether a class of finite graphs well-quasi-ordered by the
induced subgraph relation when the vertices are labelled using a finite set. In
this process, we answer positively to a conjecture of Pouzet, under the extra
assumption that the class is of bounded linear clique-width. As a byproduct of
our approach, we obtain a new proof of an earlier result from Daliagault, Rao,
and Thomass\'e, by uncovering a connection between well-quasi-orderings on
graphs and the gap embedding relation of Dershowitz and Tzameret.
