Linkless Embedding

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph have nonzero linking number. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk that is not crossed by any other feature of the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs.

Flat embeddings are automatically linkless, but not vice versa. The complete graph K6, the Petersen graph, and the other five graphs in the Petersen family do not have linkless embeddings. The linklessly embeddable graphs are closed under graph minors and Y-Δ transforms, have the Petersen family graphs as their forbidden minors, and include the planar graphs and apex graphs. They may be recognized, and a flat embedding may be constructed for them, in linear time.

Read more about Linkless Embedding:  Definitions, Examples and Counterexamples, Characterization and Recognition, Related Families of Graphs, History