Clique-width and induced topological minors
For graph theorists studying structural graph classes, this provides a complete classification for a fundamental width parameter under induced topological minor restrictions.
The paper characterizes when forbidding an induced subdivision of a graph H results in a class of bounded clique-width, showing it happens exactly when H is an induced subgraph of P4, the paw, or the diamond. This resolves an open question.
A $P_4$ is a chordless path on four vertices. A diamond is a graph obtained from a clique of size four by removing one edge of the clique. A paw is a graph obtained from a clique of size four by removing two adjacent edges of the clique. We prove that for a graph $H$, the class of graphs with no induced subdivision of $H$ has bounded clique-width if and only if $H$ is an induced subgraph of $P_4$, the paw, or the diamond. This answers a~question of Dabrowski, Johnson, and Paulusma.