## Abstract

A graph is planar if it has a drawing in which no two edges cross. The Hanani-Tutte Theorem states that a graph is planar if it has a drawing D such that any two edges in D cross an even number of times. A graph G is a non-separating planar graph if it has a drawing D such that (1) edges do not cross in D, and (2) for any cycle C and any two vertices u and v that are not in C, u and v are on the same side of C in D. Non-separating planar graphs are closed under taking minors and hence have a finite forbidden minor characterisation. In this paper, we prove a Hanani-Tutte type theorem for non-separating planar graphs. We use this theorem to prove a stronger version of the strong Hanani-Tutte Theorem for planar graphs, namely that a graph is planar if it has a drawing in which any two disjoint edges cross an even number of times or it has a chordless cycle that enables a suitable decomposition of the graph.

Original language | English |
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Article number | P1.43 |

Number of pages | 16 |

Journal | Electronic Journal of Combinatorics |

Volume | 28 |

Issue number | 1 |

DOIs | |

Publication status | Published - 26 Feb 2021 |