On Large Induced Outerplanar Subgraphs in $2$-Outerplanar Graphs

TL;DR

This paper revisits a result on large induced outerplanar subgraphs in 2-outerplanar graphs, correcting a flaw in previous proof and providing a new, more complex proof to establish the same bound.

Contribution

The paper identifies a flaw in the original proof and offers a new, more intricate proof to confirm the existence of large induced outerplanar subgraphs.

Findings

Confirmed that at least 2/3 of vertices induce an outerplanar subgraph in 2-outerplanar graphs

Provided a corrected and more complex proof of the original result

Clarified the conditions under which large induced outerplanar subgraphs exist

Abstract

Borradaile, Le and Sherman-Bennett [Graphs and Combinatorics, 2017] proved that every $n$-vertex $2$-outerplane graph has a set of at least $2n/3$ vertices that induces an outerplane graph. We identify a major flaw in their proof and recover their result with a different, and unfortunately much more complex, proof.