Non-Separating Planar Graphs

TLDR: The characterisation of non-separating planar graphs is used 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 graph in 1983.

Abstract: 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.