Connectivity is a classical indicator to reflect the reliability of network. According to the distribution of fault processors, conditional connectivity of network has received wide attention. In this paper, we prove that every minimum 3-extra-cut set \(F\) of \(r\) -regular bipartite hierarchical graph \(G\) is isomorphic to \(N_G(C_4), N_G(P_4)\) or \(N_G(S_4)\) , if for each subgraph \(G_i\) , \(G_i-F_i\) has no isolated vertices, and for a component \(H\) of \(G_1-F_1\) , which is also a component of \(G-F\) , satisfies \(|N_{G_1}(H)| \geq \alpha - 4l - |H| + 4\) and \(|N_{G - G_1}(H)| \geq l|H|\) . It follows that \(3\) -extra-connectivity \(\kappa_3(G) = |F|= min\{|N_G(C_4)|, |N_G(P_4)|, |N_G(S_4)|\}\) . As application, we derive exact values of \(3\) -extra-connectivity for modified bubble-sort graphs \(MB_n\) , star graphs \(CS_n\) , cactus-based graphs \(CN_{2n+1}\) and \(k\) -ary \(n\) -cube \(Q_n^k\) are \(4n-9\) , \(4n-10\) , \(12n-10\) and \(8n-9\) , respectively. This work generalizes the study of \(g\) -extra-connectivity from \(g\in \{1,2\}\) to \(g=3\) , extending and unifying earlier work of Gu et al. (Inf. Process Lett. 114:486-491, 2014), Hu et al. (Available at SSRN 5193978), Liu et al. (Theor. Comput. Sci. 888:95-107, 2021) and Zhu et al. (IEEE:1–6, 2017).
