[논문 리뷰] On Large Induced Outerplanar Subgraphs in $2$-Outerplanar Graphs

Marco D’Elia, Fabrizio Frati|arXiv (Cornell University)|2026. 02. 20.

Advanced Graph Theory Research인용 수 0

한 줄 요약

본 논문은 이전 증명의 결함을 수정하고, 모든 n-정점 2-outerplane 그래프가 최소 2n/3 정점의 유도된 outerplanar 부분그래프를 가진다는 것을 증명하는 새로운 귀납 알고리즘을 제시한다.

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.

연구 동기 및 목표

• Motivate and determine large induced outerplanar subgraphs within 2-outerplanar graphs as a route toward Albertson–Berman-type forest results.
• Identify and repair a major flaw in existing proofs asserting large induced outerplanar subgraphs in 2-outerplane graphs.
• Provide a self-contained, rigorous constructive proof yielding a 2n/3-sized outerplane induced subgraph in any n-vertex 2-outerplane graph.

제안 방법

• Review and formalize relevant notions for 2-outerplane graphs, outerplane embeddings, and induced subgraphs.
• Demonstrate the flaw in the previous algorithm that claimed 2n/3 bound by constructing counterexamples.
• Develop a new inductive algorithm that, given an internally-triangulated 2-outerplane graph, builds a large good set I with |I| ≥ 2n/3 such that G[I] is outerplane.
• Introduce a detailed case analysis around terminal components of G[L2] and their cage graphs, with a rooted block-cutvertex tree framework to guide recursive reductions.
• Prove that in all cases the induction can be carried through, ensuring the resulting subgraph remains outerplane.

Figure 1 : Illustration for the correctness of Step 4 in the proof of Lemma ˜ 2.1 . The cycle $\mathcal{C}$ is represented by a thick line and the face $f$ is shaded orange.

실험 결과

연구 질문

• RQ1 Can every n-vertex 2-outerplane graph contain an induced outerplane subgraph on at least 2n/3 vertices?
• RQ2 Where did the prior proof by Borradaile, Le and Sherman-Bennett fail, and how can a corrected argument be constructed?
• RQ3 What structural decompositions (terminal components, cage graphs, and rooted block-cutvertex trees) enable a correct inductive proof?
• RQ4 Does the corrected approach yield a polynomial-time algorithm to compute the large induced outerplanar subgraph?
• RQ5 How do auxiliary constructs (e.g., pesky vs cushy blocks) influence the inductive step and stopping conditions?

주요 결과

• A corrected proof establishes that every n-vertex 2-outerplane graph contains a set I of size at least 2n/3 whose induced subgraph is outerplane.
• The paper identifies and explains a major flaw in the prior proof and demonstrates counterexamples where the previous approach fails.
• An inductive algorithm with a comprehensive case distinction around terminal components of G[L2] and cage graphs is developed to achieve the 2n/3 bound.
• The construction ensures G[I] remains outerplane at each step, leading to a polynomial-time implementable procedure (outline provided).
• The work clarifies the boundary between G[L2] and L1 structures and uses a rooted dual-tree and block-cutvertex trees to guide reductions.

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