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Projective Planar Graph


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 >=2 include the complete graph K_6 and Petersen graph P.

ProjectivePlaneCrossingNumberForbiddenMinors

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 E_(11) (Hlinenỳ 2010; Shahmirzadi 2012, p. 7, Fig. 1.1) is not isomorphic to the graph E_(11) of Glover and Huneke (1978) and Mohar and Thomassen (2011) and therefore appears to be incorrectly drawn. Note also that this set includes the graph unions 2K_(3,3) and 2K_5, 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).


See also

Forbidden Minor, Projective Plane Crossing Number

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References

Archdeacon, D. S. "A Kuratowski Theorem for the Projective Plane." Ph. D. thesis. Columbus, OH: Ohio State University, 1980.Archdeacon, D. "A Kuratowski Theorem for the Projective Plane." J. Graph Th. 5, 243-246, 1981.Glover, H.; and Huneke, J. "The Set of Irreducible Graphs for the Projective Plane Is Finite." Disc. Math. 22< 243-256, 1978.Glover, H.; Huneke, J. P.; and Wang, C. S. "103 Graphs That Are Irreducible for the Projective Plane." J. Combin. Th. Ser. B 27, 332-370, 1979.Hlinenỳ, P. "20 Years of Negami's Planar Cover Conjecture." Graphs and Combinatorics 26, 525-536, 2010.Ho, P. T. "The Crossing Number of K_(4,n) 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. https://arxiv.org/abs/2002.00258.Mohar, B, and Thomassen, C. "The Minimal Forbidden Sugraphs for the Projective Plane." Appendix A in Graphs on Surfaces. Baltimore, MD: Johns Hopkins University Press, pp. 247-252, 2001.Shahmirzadi, A. S. "Minor-Minimal Non-Projective Planar Graphs with an Internal 3-Separation." Ph.D. thesis. Atlanta, GA: Georgia Institute of Technology, Dec. 2012.

Cite this as:

Weisstein, Eric W. "Projective Planar Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProjectivePlanarGraph.html

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