Six variations on a theme: almost planar graphs

Abstract

A graph is apex if it can be made planar by deleting a vertex, that is, there exists $v$ such that $G-v$ is planar. We also define several related notions; a graph is edge apex if there exists $e$ such that $G-e$ is planar, and contraction apex if there exists $e$ such that $G∕e$ is planar. Additionally we define the analogues with a universal quantifier: for all $v$, $G-v$ is planar; for all $e$, $G-e$ is planar; and for all $e$, $G∕e$ is planar. The graph minor theorem of Robertson and Seymour ensures that each of these six notions 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 (there exists $e$ such that $G+e$ is nonplanar, and for all $e$, $G+e$ is nonplanar) and determine the corresponding minor minimal graphs.