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.
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).
