# Non-separating Planar Graphs

## TL;DR

This paper characterizes non-separating planar graphs through forbidden minors and structural properties, and applies this to identify maximal linkless graphs with specific edge counts, answering a longstanding question.

## Contribution

It provides a complete minor-based characterization and structural description of non-separating planar graphs, and demonstrates their application in graph theory.

## Key findings

- Characterization of non-separating planar graphs via forbidden minors.
- Structural description of maximal non-separating planar graphs.
- Existence of maximal linkless graphs with 3n-3 edges.

## 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.