“…Such a classification already exists for H-free graphs, as observed in [22]: for a graph H, the class of H-free graphs has bounded linear rank-width if and only if H is a subgraph of P 3 not isomorphic to 3P 1 . We note that similar classifications also exist for other width parameters: for the tree-width of pH 1 , H 2 q-free graphs [7], which was later generalized to a classification for tree-width of H-free graphs, where H is a finite set of graphs [37], rank-width of H-free graphs (see [24]), rank-width of H-free bipartite graphs [23,36,38], and up to five non-equivalent open cases, rank-width of pH 1 , H 2 q-free graphs (see [8] or [22]), and for the mim-width of H-free graphs [13], whereas there is still an infinite number of open cases left for the mim-width of pH 1 , H 2 q-free graphs [13].…”