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

2026-02-20

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.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• outerplanar graph
• 2-outerplanar graph
• induced subgraph
• graph theory

Innovation

• 2-outerplanar graphs
• outerplanar subgraphs
• induced subgraph
• graph theory
• proof correction