Editing Linkless embedding (section)

=== Apex graphs ===

[[File:Apex rhombic dodecahedron.svg|thumb|A linkless apex graph that is not YΔY reducible.]]
The planar graphs and the [[apex graph]]s are linklessly embeddable, as are the graphs obtained by [[YΔ- and ΔY-transformation]]s from these graphs.<ref name="rst93a"/> The ''YΔY reducible graphs'' are the graphs that can be reduced to a single vertex by YΔ- and ΔY-transformations, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices; they are also minor-closed, and include all planar graphs. However, there exist linkless graphs that are not YΔY reducible, such as the apex graph formed by connecting an apex vertex to every degree-three vertex of a [[rhombic dodecahedron]].<ref>{{harvtxt|Truemper|1992}}.</ref> There also exist linkless graphs that cannot be transformed into an apex graph by YΔ- and ΔY-transformation, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices: for instance, the ten-vertex [[crown graph]] has a linkless embedding, but cannot be transformed into an apex graph in this way.<ref name="rst93a"/>