An apex graph is a graph possessing at least one graph vertex whose removal results in a planar graph. The set of graph vertices whose removal results in a planar graph is known as the apices of the graph.
Planar graphs are therefore 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 graph 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 graph vertex whose removal gives a planar graph, while a critical nonplanar graph requires that removal of each graph 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 
.
Apex graphs have Hadwiger number at most five since they include the complete graph 
among their forbidden minors.
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 graph
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 graph 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).
Inoue et al. (2026) presented a computer-assisted proof that every biconnected cubic graph that is apex has edge chromatic number 3. Since snarks are biconnected cubic graphs with edge chromatic number 4, the claimed result implies that no snark is apex and supplies the remaining apex case of Tutte three-edge-coloring conjecture.
