Structure fault tolerance of k-ary n-cube networks

Let G be a graph and H be a certain connected subgraph of G. The H-structure connectivity κ(G;H) (resp. H-substructure connectivity κ^s(G;H)) of G is the minimum number of a set of subgraphs F={H₁,H₂,⋯,H_m} (resp. F={H₁ ^′,H₂ ^′,⋯,H_m ^′}) such that H_i is isomorphic to H (resp. H_i ^′ is a connected subgraph of H) for every 1≤i≤m, and F's removal will disconnect G. For the k-ary n-cube Q_n ^k, the κ(Q_n ^k;H) and κ^s(Q_n ^k;H) were determined for H∈{K₁,K_1,1,K_1,2,K_1,3}. In this paper, we show κ(Q_n ^k;H) and κ^s(Q_n ^k;H) for H∈{P_l,C_l} where 3≤l≤2n.