P_k and C_k structure and substructure connectivity of hypercubes

Hypercube is one of the most important networks to interconnect processors in multiprocessor computer systems. Different kinds of connectivities are important parameters to measure the fault tolerability of networks. Lin et al.\cite{LinStructure} introduced the concept of $H$-structure connectivity $\kappa (Q_n;H)$ (resp. $H$-substructure connectivity ${\kappa }^s(Q_n;H)$) as the minimum cardinality of $F=\{H_1,\dots ,H_m\}$ such that $H_i(i=1,\dots ,m)$ is isomorphic to $H$ (resp. $F=\{H_1^{\mathrm{\prime }},\dots ,H_m^{\mathrm{\prime }}\}$ such that $H_i^{\mathrm{\prime }}(i=1,\dots ,m)$ is isomorphic to connected subgraphs of $H$) such that $Q_n-V(F)$ is disconnected or trivial. In this paper, we discuss $\kappa (Q_n;H)$ and ${\kappa }^s(Q_n;H)$ for hypercubes $Q_n$ with $n\ge 3$ and $H\in {P_k,C_k|3\leq k\leq 2^{n-1}}$. As a by-product, we solve the problem mentioned in \cite{ManeStructure}.