The study of conditional connectivity has certain limitations in characterizing fault tolerance, as it often fails to fully account for the structural relationship between adjacent processors in multiprocessor systems. To address this issue, two indicators, namely the R-structure connectivity and the R-substructure connectivity, have been proposed in the literature to better measure the fault tolerance of interconnection networks. Let G be a connected graph, and R, M be two connected subgraphs of G. The R-structure (resp. R-substructure) connectivity, denoted by \(\kappa (G;R)\) (resp. \(\kappa ^{s}(G;R)\) ), is the cardinality of a minimum set of subgraphs \(\mathcal {F}=\{F_{1},\cdots ,F_{t}\}\) in G such that every \(F_{i}\in \mathcal {F}\) is isomorphic to R (resp. to a connected subgraph of R) and \(G-F\) is disconnected. In particular, we call \(\mathcal {F}\) a minimum R-structure (resp. R-substructure) cut of G. Moreover, G is called hyper (sub-) \(R|_{M}\) -connected if the removal of every minimum R-structure (R-substructure) cut divides G into exactly two components, one of which is isomorphic to M. In this paper, the R-structure connectivity and the R-substructure connectivity of hierarchical pancake networks \(HPG_{n}\) for \(n\ge 3\) are studied for the first time, where \(R\in \{K_{1,r}\mid 1\le r\le n-1\}\) . Furthermore, we prove that \(HPG_{n}\) is both hyper \(R|_{K_{1}}\) -connected and hyper sub- \(R|_{K_{1}}\) -connected for \(n\ge 4.\)
