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An intrinsically linked graph is a graph having the property that any embedding of it in three dimensions contains a nontrivial link. A graph is intrinsically linked iff it contains one of the seven Petersen family graphs (Robertson et al. 1993) as a graph minor.

The complete graph (left) is intrinsically linked because it contains at least two linked triangles. The complete k-partite graph (right) is also intrinsically linked.

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

A graph on nodes with edge count is intrinsically linked.

Complete Graph, Complete k-Partite Graph, Linklessly Embeddable 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.

## Cite this as:

Weisstein, Eric W. "Intrinsically Linked Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntrinsicallyLinkedGraph.html