Graph Minor

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

The Kuratowski reduction theorem states that any nonplanar graph has the complete graph  or the complete bipartite graph  as a minor. In addition, any snark has the Petersen graph as a minor, as conjectured by Tutte (1967; West 2000, p. 304) and proved by Robertson et al.

The determination of graph minors is an NP-hard problem for which no good algorithms are known, although brute-force methods such as those due to Robertson, Sanders, and Thomas exist.

For any fixed graph , it is possible to test whether  is a minor of an 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.

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: 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.; and Thomas, R. "The Four-Color Theorem." http://www.math.gatech.edu/~thomas/FC/fourcolor.html.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.