The Kuratowski-edge conjecture for crossing-critical graphs

Let $G$ be a graph. An edge of $G$ is a Kuratowski edge if it belongs to a subgraph homeomorphic to $K_{3,3}$ or $K_5$. Kuratowski-edge conjecture. If every edge of $G$ is a Kuratowski edge, then $G$ can be made crossing-critical by a suitable multiplication of its edges.

The conjecture proposes a structural sufficient condition for a graph to admit an edge multiplication that makes it crossing-critical. The paper establishes positive results for a large family of graphs and for individual Kuratowski edges, but notes that a full characterization of the relevant class remains open.