A graph with projective plane crossing number equal to 0 may be said to be projective planar. Examples of projective
planar graphs with graph crossing number include the complete
graph
and Petersen graph
.
Embeddability in the projective plane (i.e., graphs with projective plane crossing number 0) are characterized by a set of exactly 35 forbidden
minors (Glover et al. 1979; Archdeacon 1981; Hlinenỳ 2010; Shahmirzadi
2012, p. 7, Fig. 1.1). Note that this set includes the graph unions and
, each member of which is embeddable in the projective plane.
This means that, unlike planar graphs, disjoint unions of graphs which are embeddable
in the projective plane may not themselves be embeddable. As of 2022, the plane and
projective plane are the only surfaces for which a complete list of forbidden minors
is known (Mohar and Škoda 2020).
Richter and Siran (1996) computed the crossing number of the complete bipartite graph on an arbitrary surface. Ho (2005) showed that the projective
plane crossing number of
is given by
For ,
2, ..., the first few values are therefore 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, ...
(OEIS A128422).