  A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$
on the plane such that (1) no two edges cross each other in $D$ and (2) for any
cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in
$D$.
  Non-separating planar graphs are closed under taking minors and are a
subclass of planar graphs and a superclass of outerplanar graphs.
  In this paper, we show that a graph is a non-separating planar graph if and
only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$
as a minor.
  Furthermore, we provide a structural characterisation of this class of
graphs. More specifically, we show that any maximal non-separating planar graph
is either an outerplanar graph or a subgraph of a wheel or it can be obtained
by subdividing some of the side-edges of the 1-skeleton of a triangular prism
(two disjoint triangles linked by a perfect matching).
  Lastly, to demonstrate an application of non-separating planar graphs, we use
the characterisation of non-separating planar graphs to prove that there are
maximal linkless graphs with $3n-3$ edges which provides an answer to a
question asked by Horst Sachs about the number of edges of linkless graphs in
1983.
