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.

Highlights

  • The Hanani-Tutte Theorem is a well-known theorem about planar graphs: Strong Hanani-Tutte Theorem ([12])

  • A drawing D of a graph G in a plane Σ partitions all the points of Σ\D into a set of regions, denoted by regions(D), such that any two points p and q are in the same region r ∈ regions(D) if and only if there is a curve from p to q that does not cross any vertex or edge of D

  • Graph G is a dual of an Eulerian planar graph and it is bipartite

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Summary

Introduction

The Hanani-Tutte Theorem is a well-known theorem about planar graphs: Strong Hanani-Tutte Theorem ([12]). In this paper we first prove a Hanani-Tutte type theorem for non-separating planar graphs. Let D be a non-separating drawing of a graph G such that any two vertexdisjoint edges in D cross each other an even number of times. Since a drawing in which any two vertex-disjoint edges cross each other an even number of times is evenly decomposable, Theorem 2 is at least as strong as the strong version of the Hanani-Tutte Theorem. Since there are evenly decomposable drawings in which there is a pair of vertex-disjoint edges that cross each other an odd number of times, Theorem 2 is stronger than the strong version of the Hanani-Tutte Theorem.

Background
Preliminary Results
Hanani-Tutte and Non-separating Planar Graphs
Conclusion
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