The \(FTCQ_{n}\) , an n-dimensional folded twisted crossed cube proposed by Guo, is derived from the twisted crossed cube \(TCQ_n\) by including an additional set of \(2^{n-1}\) edges. The h-extra connectivity of a connected graph G, denoted by \(\kappa _{h}(G)\) , is the minimum number of vertices that need to be removed in order to disconnect G, while ensuring that the remaining graph contains no fewer than \(h+1\) vertices. The h-extra diagnosability of a graph G requires that each fault-free component of G must contain no fewer than \(h+1\) vertices. The h-extra diagnosability of a graph G is denoted as \(t_{h}^{PMC}(G)\) under the PMC model and as \(t_{h}^{MM^*}(G)\) under the MM \(^*\) model. In this paper, we determine the h-extra connectivity of \(TCQ_{n}\) is \(\kappa _{h}(TCQ_{n})=n(h+1)-\frac{1}{2}h^{2}-\frac{3}{2}h\) for \(n\ge 5\) , \(0\le h \le \lfloor \frac{n}{2} \rfloor\) , and the h-extra diagnosability of \(TCQ_{n}\) is \(t_{h}^{PMC}(TCQ_{n})=(n+1)h-\frac{1}{2}h^{2}-\frac{1}{2}h\) for \(n\ge 5\) , \(0\le h \le \lfloor \frac{n}{2} \rfloor\) , under the PMC model, and \(t_{h}^{MM^*}(TCQ_{n})=n(h+1)-\frac{1}{2}h^{2}-\frac{1}{2}h\) for \(n \ge 6\) , \(0 \le h\le \frac{1}{2}(n-2)\) under the MM \(^{*}\) model. Furthermore, we investigate the h-extra connectivity of \(FTCQ_{n+1}\) is \(\kappa _{h}(FTCQ_{n+1})=(n+2)(h+1)-\frac{1}{2}h^{2}-\frac{3}{2}h\) for \(n \ge 7\) , \(0\le h \le \lfloor \frac{n}{2}\rfloor\) , and the h-extra diagnosability of \(FTCQ_{n+1}\) is \(t_{h}^{PMC}(FTCQ_{n+1})=(n+2)(h+1)-\frac{1}{2}h^{2}-\frac{1}{2}h\) for \(n\ge 7\) , \(0\le h \le \lfloor \frac{n}{2} \rfloor\) , under the PMC model.
