Six variations on a theme: almost planar graphs

TL;DR

This paper explores variations of almost planar graphs, identifying finite obstruction sets for some definitions and providing bounds for others, advancing understanding of near-planar graph classes.

Contribution

It determines the finite obstruction sets for universal apex properties and provides bounds for apex, edge apex, and contraction apex graphs, along with approaches for almost nonplanar graphs.

Findings

Finite obstruction sets identified for universal apex properties.

Lower bounds established for apex, edge apex, and contraction apex obstructions.

Methods developed for almost nonplanar graphs.

Abstract

A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that $G/e$ is planar, as well as the analogues with a universal quantifier: $\forall v$, $G-v$ planar; $\forall e$, $G-e$ planar; and $\forall e$, $G/e$ planar. The Graph Minor Theorem of Robertson and Seymour ensures that each of these six gives rise to a finite set of obstruction graphs. For the three definitions with universal quantifiers we determine this set. For the remaining properties, apex, edge apex, and contraction apex, we show there are at least 36, 55, and 82 obstruction graphs respectively. We give two similar approaches to almost nonplanar ($\exists e$, $G+e$ is nonplanar and $\forall e$, $G+e$ is nonplanar) and determine the corresponding minor minimal graphs.