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 the graph 
(Hlinenỳ 2010; Shahmirzadi 2012, p. 7, Fig. 1.1)
is not isomorphic to the graph 
of Glover and Huneke (1978) and Mohar and Thomassen (2011)
and therefore that figure gives an incorrect drawing of 
. In the Wolfram
Language, the corrected graph is implemented as GraphData[
"ProjectivePlanarForbiddenMinor",
16
],
while the nonisomorphic drawing is preserved as GraphData[
"IncorrectProjectivePlanarForbiddenMinor",
16
].
Note also 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).
There are exactly 103 projective planar forbidden subgraphs (Glover et al. 1979; Archdeacon 1980, 1981; Mohar and Thomassen 2001).
on the Real Projective Plane." Disc. Math. 304,
23-33, 2005.Mohar, B. and Škoda, P. "Excluded Minors for
the Klein Bottle I. Low Connectivity Case." 1 Feb 2020.