Minimal obstructions for skew- and X-augmentations

List the minimal forbidden induced subgraphs for skew-augmentations of line graphs of bipartite multigraphs and for X-augmentations of line graphs of incidence graphs.

Background

The paper gives structural characterizations of skew-augmentations of line graphs of bipartite multigraphs and X-augmentations of line graphs of incidence graphs through forbidden 2-edge-coloured patterns. For most characterized classes, the authors also provide lists of minimal graph obstructions. They explicitly identify the corresponding obstruction descriptions for these two augmentation classes as unresolved.

References

However, we left two such characterizations open: list the minimal obstructions of skew-augmentations of bipartite multigraphs, and of X-augmentations of line graphs of incidence graphs (compare to Theorems 22 and 23).

— On the expressive power of $2$-edge-colourings of graphs  (2503.07409 - Bok et al., 10 Mar 2025) in Section 10, Conclusions; comparison with Theorems 22 and 23