Six variations on a theme: almost planar graphs
Max Lipton, Eoin Mackall, Thomas W. Mattman, Mike Pierce, Samantha, Robinson, Jeremy Thomas, and Ilan Weinschelbaum

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, such that is planar. We define the related notions of edge apex, such that is planar, and contraction apex, such that is planar, as well as the analogues with a universal quantifier: , planar; , planar; and , 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 (, is nonplanar and , is nonplanar) and determine the corresponding…
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