Characterization of class-2 projective planar r-graphs for r>3

Characterize all class-2 projective planar r-graphs for integers r>3, including determining whether constructions other than the described Petersen-based constructions exist.

Background

The paper defines Petersen-based constructions, called P^M-like r-graphs, and shows that every such graph is class 2. It also exhibits infinitely many projective planar 4-graphs of class 2 that are not P^M-like, demonstrating that these constructions do not provide a complete classification even when r=4. The authors therefore identify the complete characterization for r>3 as unresolved and explicitly note the possibility of further constructions.