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F\"uredi and Gunderson showed that ex (n, C₂₊+₁) is achieved only on K₂, {2} if n 4k-2. It is natural to study how far a C₂₊+₁-free graph is from being bipartite. Let T^* (r, n) be obtained by adding a suspension Kₑ with 1 suspension point to K-ₑ+₁₂, -r+1{2}. We show that for integers r, k with 3 r 2k-4 and n 20 (r+2) ²k, if G is a C₂₊+₁-free n-vertex graph with e (G) e (T^* (r, n) ), then G is obtained by adding suspensions to a bipartite graph one by one and the total number of vertices in all suspensions minus intersection points is no more than r-1. In other words, G=B₈=₁ᵖ Gᵢ, where B is a bipartite graph, G₁ is a suspension to B, Gⱼ is a suspension to B₈=₁^j-1 Gᵢ for 2 j p and ₈=₁ᵖ V (Gᵢ) -V (Gᵢ) V (B₈=₁^j-1 Gᵢ) r-1. Furthermore, ₈=₁ᵖ V (Gᵢ) -V (Gᵢ) V (B₈=₁^j-1 Gᵢ) = r-1 if and only if G=T^* (r, n). Let d₂ (G) =\|T|: T V (G), G-T \ is bipartite\ and ₂ (G) =\|E|: E E (G), G-E \ is bipartite\. Our structural stability result implies that d₂ (G) r-1 and ₂ (G) {2 2}+{2 2} under the same condition, which is a recent result of Ren-Wang-Wang-Yang SIAM J. Discrete Math. 38 (2024). They proved d₂ (G) r-1 and ₂ (G) {2 2}+{2 2} separately. We introduce a new concept strong-2k-core which is the key that we can give a stronger structural stability result but a simpler proof.
Yan et al. (Tue,) studied this question.