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.
Every graph minor is a topological minor, but
the converse does not necessarily hold.
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.
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. ACM35, 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. B63, 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.
is a minor of a graph 
if a copy of 
can be obtained from 
via repeated edge deletion and/or edge contraction.
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.
,
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).
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.
