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Linklessly Embeddable Graph

A linklessly embeddable graph is a graph having the property that there exists an embedding in three dimensions that does not contain a nontrivial link. A graph is linklessly embeddable iff it does not contain one of the seven Petersen family graphs as a forbidden minor (Robertson et al. 1993).

Apex graphs (and therefore planar graphs) are linklessly embeddable.

Linklessly embeddable graphs have Hadwiger number at most five since they include the complete graph K_6 among their forbidden minors (Robertson et al. 1993).

A graph that is not linklessly embeddable is said to be an intrinsically linked graph.

See also

Intrinsically Linked Graph, Petersen Family Graphs

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References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 217-221, 1994.Naimi, R.; Pavelescu, A.; and Pavelescu, E. "New Bounds of Maximal Linkless Graphs." 20 Sep 2020. https://arxiv.org/abs/2007.10522.Odeneal, Y.; Naimi, R.; Pavelescu, A.; and Pavelescu, E. "The Complement Problem for Linklessly Embeddable Graphs." J. Knot Theory and Its Ramifications 2250075, 1-10, 2022.Robertson, N.; Seymour, P. D.; and Thomas, R. "Linkless Embeddings of Graphs in 3-Space." Bull. Amer. Math. Soc. 28, 84-89, 1993.

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Linklessly Embeddable Graph

Cite this as:

Weisstein, Eric W. "Linklessly Embeddable Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LinklesslyEmbeddableGraph.html

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