A planar graph
is said to be nonseparating if there exists an embedding of
in
such that, for any
cycle
, all vertices of
are within the same
connected component of
.
Dehkordi and Farr classified the nonseparating planar graphs as either outerplanar graphs, subgraphs of
wheel graphs, or subgraphs of elongated triangular prisms. We use maximal nonseparating planar graphs to
construct examples of maximal linkless graphs and maximal knotless graphs. We show that, for a maximal
nonseparating planar graph
with
vertices,
the complement
is
–apex.
This implies that the Colin de Verdière invariant of the complement
satisfies
. We show this
to be an equality. As a consequence, the conjecture of Kotlov, Lovász and Vempala that, for
a simple graph
,
is true for
–apex graphs
for which
is planar
nonseparating. It also follows that complements of nonseparating planar graphs of order at
least nine are intrinsically linked. We prove that the complements of nonseparating planar
graphs
of order at least ten are intrinsically knotted.
Keywords
nonseparating planar graph, Colin de Verdière invariant,
intrinsically linked graph, intrinsically knotted graph
