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.

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.

By the Robertson-Seymour theorem, since apex graphs are closed under minors, they have a finite obstruction set consisting
of forbidden minors. The forbidden minors of apex
graphs include the Petersen family graphs
and the disjoint unions of ,
, and (Pierce 2014, p. 8; Thomas 2014). Pierce (2014)
identified 157 minimal apex-forbidden minors, but a complete list of forbidden minors
is not yet known (Gupta and Impagliazzo 1991, Pierce 2014).

Gupta, A. and Impagliazzo, R. "Computing Planar Intertwines." In Proc. 32nd IEEE Symposium on Foundations of Computer Science (FOCS '91).
IEEE Computer Society, pp. 802-811, 1991.Pierce, M. "Searching
for and Classifying the Finite Set of Minor-Minimal Non-Apex Graphs." Honors
thesis. Chico, CA: California State University, Chico, 2014. http://tmattman.yourweb.csuchico.edu/mpthesis.pdf.Sloane,
N. J. A. Sequences A215620 and A215621 in "The On-Line Encyclopedia of Integer
Sequences."Thomas, J. "Properties of 2-Connected MMNA Graphs."
Preprint, 2014.