Apex Graph

An apex graph is a graph possessing at least one vertex whose removal results in a planar graph. The set of vertices whose removal results in a planar graph is known as the apices of the graph.

Planar graphs are threfore trivial apex graphs having all vertices as apices.

A nonplanar apex graph, sometimes also called a nearly planar graph (though this term is also used in other contexts), is a nonplanar graph possessing at least one vertex whose removal results in a planar graph.

Apex graphs differ from critical nonplanar graphs in that an apex graph requires only that there exist at least one vertex whose removal gives a planar graph, while a critical nonplanar graph requires that removal of each vertex gives a planar graph.

There exist nonplanar apex graphs that are not singlecross graphs. For example, the complete bipartite graph  is nonplanar and apex but has graph crossing number 2.

Every apex graph has chromatic number .

The numbers of apex graphs on , 2, ... vertices are 1, 2, 4, 11, 34, 155, 1026, 11666, 226916, ... (OEIS A215620), while the numbers of nonplanar apex graphs are 0, 0, 0, 0, 1, 13, 204, 4700, 147063, ... (OEIS A215621), with the only nonplanar apex graph on  vertices being the complete graph . The 13 nonplanar apex graphs on six vertices are illustrated above. In these graphs, the apices consist of all but 1 and 2 for the -Turán graph, all but vertex 2 for the , , , and -lollipop graphs, and all vertices for each of the others.

REFERENCES

Sloane, N. J. A. Sequences A215620 and A215621 in "The On-Line Encyclopedia of Integer Sequences."