Editing Linkless embedding (section)

==Examples and counterexamples==
[[File:Petersen family.svg|thumb|The [[Petersen family]].]]
As {{harvtxt|Sachs|1983}} showed, each of the seven graphs of the Petersen family is intrinsically linked: no matter how each of these graphs is embedded in space, they have two cycles that are linked to each other. These graphs include the [[complete graph]] ''K''<sub>6</sub>, the [[Petersen graph]], the graph formed by removing an edge from the [[complete bipartite graph]] ''K''<sub>4,4</sub>, and the complete tripartite graph ''K''<sub>3,3,1</sub>.

Every [[planar graph]] has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space. If a graph is planar, this is the only way to embed it flatly and linklessly into space: every flat embedding can be continuously deformed to lie on a flat plane. And conversely, every nonplanar linkless graph has multiple linkless embeddings.<ref name="rst93a"/>

[[File:Apex graph.svg|thumb|An [[apex graph]]. If the planar part of the graph is embedded on a flat plane in space, and the apex vertex is placed above the plane and connected to it by straight line segments, the resulting embedding is flat.]]
An [[apex graph]], formed by adding a single vertex to a planar graph, also has a flat and linkless embedding: embed the planar part of the graph on a plane, place the apex above the plane, and draw the edges from the apex to its neighbors as line segments. Any closed curve within the plane bounds a disk below the plane that does not pass through any other graph feature, and any closed curve through the apex bounds a disk above the plane that does not pass through any other graph feature.<ref name="rst93a"/>

If a graph has a linkless or flat embedding, then modifying the graph by subdividing or unsubdividing its edges, adding or removing multiple edges between the same pair of points, and performing [[YΔ- and ΔY-transformation]]s that replace a degree-three vertex by a triangle connecting its three neighbors or the reverse all preserve flatness and linklessness.<ref name="rst93a"/> In particular, in a [[cubic graph|cubic]] planar graph (one in which all vertices have exactly three neighbors, such as the cube) it is possible to make duplicates of any [[Independent set (graph theory)|independent set]] of vertices by performing a YΔ-transformation, adding multiple copies of the resulting triangle edges, and then performing the reverse ΔY-transformations.