Dean \(([3])\) shows that if \(G\) be a \(k\)-connected graph such that any fragment whose neighborhood contains an edge has cardinality exceeding \(\frac{k}{2}\), then the subgraph \(H = (V(G), E_k(G))\) formed by \(V(G)\) and the \(k\)-contractible edges of \(G\) is \(2\)-connected. In this paper, we show that for \(k = 4\), Dean’s result holds when reduced \(\frac{k}{2}\) to \(\frac{k}{4}\). But for \(k \geq 5\), we give a counterexample to show that it is false and give a lower bound of the number of \(k\)-contractible edges for \(k = 5\).
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