🤖 AI Summary
This study addresses the problem of establishing a lower bound on the number of vertices in a maximum induced outerplanar subgraph of a 2-outerplanar graph. In response to a critical flaw identified in prior proofs, we introduce a novel proof framework grounded in structural graph analysis and inductive construction, leveraging the embedding properties of outerplanar graphs to ensure rigorous derivation. Our approach not only rectifies the shortcomings of earlier arguments but also formally establishes that every 2-outerplanar graph with $n$ vertices contains an induced outerplanar subgraph on at least $2n/3$ vertices. This result reaffirms and strengthens the theoretical foundation of this classical lower bound.
📝 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.
