A graph minor of a graph is a graph for which a copy of can be obtained from via repeated edge deletion, vertex deletion, and/or edge contraction.

Wagner's theorem states that a graph is nonplanar iff it has the complete graph or the complete bipartite graph as a minor. Tutte three-edge-coloring conjecture asserts that every snark has the Petersen graph as a minor.

For any fixed graph , it is possible to test whether is a minor of a given graph in polynomial time, so if a forbidden minor characterization is available, then any graph property which is preserved by deletions and contractions may be recognized in polynomial time (Fellows and Langston 1988, Robertson and Seymour 1995).

As of 2022, the plane and projective plane are the only surfaces for which a complete list of forbidden minors is known for graph embedding (Mohar and Škoda 2020).

A graph is called a topological minor of a graph if a graph expansion of is isomorphic to a subgraph of . Every topological minor is also a minor, but the converse is not necessarily true.

See also

Choice Number, Edge Contraction, Edge Deletion, Forbidden Minor, Kuratowski's Theorem, Matroid Minor, Projective Plane Crossing Number, Robertson-Seymour Theorem, Topological Minor, Vertex Deletion, Wagner's Theorem

This entry contributed by Ed Pegg, Jr. (author's link)

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References

Demaine, E. D.; Hajiaghayi, M.; and Kawarabayashi, K.-I. "Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring." In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), Pittsburgh, PA, October 23-25, 2005. pp. 637-646.Demaine, E. D.; Hajiaghayi, M.; and Kawarabayashi, K.-I. "Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner's Contraction." In Algorithms and Computation. Proceedings of the 17th International Symposium (ISAAC 2006) Held in Kolkata, December 18-20, 2006 (Ed. T. Asano). Berlin, Germany: Springer, pp. 3-15, 2006.Fellows, M. R. and Langston, M. A. "Nonconstructive Tools for Proving Polynomial-Time Decidability." J. ACM 35, 727-739, 1988.Mohar, B. and Škoda, P. "Excluded Minors for the Klein Bottle I. Low Connectivity Case." 1 Feb 2020. https://arxiv.org/abs/2002.00258.Robertson, N.; Sanders, D. P.; Seymour, P. D.; and Thomas, R. "A New Proof of the Four Colour Theorem." Electron. Res. Announc. Amer. Math. Soc. 2, 17-25, 1996.Robertson, N.; Sanders, D. P.; Seymour, P. D.; and Thomas, R. "The Four-Colour Theorem." J. Combin. Th. Ser. B 70, 2-44, 1997. https://doi.org/10.1006/jctb.1997.1750.Robertson, N. and Seymour, P. D. "Graph Minors. XIII. The Disjoint Paths Problem." J. Combin. Th., Ser. B 63, 65-110, 1995.Tutte, W. T. "A Geometrical Version of the Four Color Problem." In Combinatorial Math. and Its Applications (Ed. R. C. Bose and T. A. Dowling). Chapel Hill, NC: University of North Carolina Press, 1967.West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000.

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Graph Minor

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Weisstein, Eric W., with contributions by Ed Pegg, Jr.. "Graph Minor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphMinor.html

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