An Improved Interpolation Theorem and Disproofs of Two Conjectures on 2-Connected Subgraphs

We prove that any $2$-connected graph $G$ on $n$ vertices with minimum degree $\delta (G)\ge \frac{n}{4}+2$ contains a $2$-connected subgraph of order $k$ for every integer $k$ with $4\le k\le n$. This improves a previous result of Yin and Wu. In \cite{YinWu-DAM-2026}, Yin and Wu proposed two conjectures. The first states that for any $2$-connected graph $G$ of order $n$ and size $m$, there exists a $2$-connected subgraph of order $k$ for each $k\in \{4,\dots ,n\}$ whenever $m\ge \frac{1}{2}{n}^{3\mathrm{/}2}$. The second conjecture asserts that the same conclusion holds under the alternative condition $\delta (G)\ge \sqrt{n}$. In this paper, we construct counterexamples that completely disprove the first conjecture. Furthermore, using the existence of $(v,k,2)$-Symmetric Balanced Incomplete Block designs (i.e., SBIBDs), we disprove the second conjecture for all $n\in \{8,14,22,32,74,112,158\}$. Finally, we propose a conjecture of our own: for any $2$-connected graph $G$ on $n$ vertices with $\delta (G)\ge \frac{n}{k}$, where $k\ge 3$ and $n$ is sufficiently large, $G$ contains a $2$-connected subgraph of every order from $4$ to $n$.