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

Borradaile, Le and Sherman-Bennett [Graphs and Combinatorics, 2017] proved that every $n$-vertex $2$-outerplane graph has a set of at least $2n\mathrm{/}3$ 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.