A Planarity Criterion for Graphs

TL;DR

This paper introduces a new combinatorial planarity criterion for graphs based on the grounding of cocycles, providing an alternative to Kuratowski's Theorem and offering insights into graph planarity characterization.

Contribution

It presents a novel planarity criterion using cocycle grounding, independent of K_{3,3} and K_5, with proofs and variants for characterizing planar graphs.

Findings

A new planarity criterion based on cocycle grounding.

Equivalence of the criterion with classical planarity conditions.

Variants of the criterion that are also necessary and sufficient.

Abstract

It is proven that a connected graph is planar if and only if all its cocycles with at least four edges are "grounded" in the graph. The notion of grounding of this planarity criterion, which is purely combinatorial, stems from the intuitive idea that with planarity there should be a linear ordering of the edges of a cocycle such that in the two subgraphs remaining after the removal of these edges there can be no crossing of disjoint paths that join the vertices of these edges. The proof given in the paper of the right-to-left direction of the equivalence is based on Kuratowski's Theorem for planarity involving K_{3,3} and K_5, but the criterion itself does not mention K_{3,3} and K_5. Some other variants of the criterion are also shown necessary and sufficient for planarity.