On almost-planar graphs

TL;DR

This paper provides a concise proof of Gubser's characterization of 3-connected almost-planar graphs and explores the structure of such graphs beyond 3-connectivity.

Contribution

It offers a shorter proof of a known characterization and analyzes the structure of almost-planar graphs that are not 3-connected.

Findings

Short proof of Gubser's 3-connected almost-planar graph characterization

Structural insights into non-3-connected almost-planar graphs

Clarification of properties distinguishing almost-planar graphs

Abstract

A nonplanar graph G is called almost-planar if for every edge e of G, at least one of G\e and G/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small portion of it was published. The main purpose of this paper is to provide a short proof of this result. We also discuss the structure of almost-planar graphs that are not 3-connected.