Structure Connectivity and Substructure Connectivity of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Ary <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-Cube Networks

We present new results on the fault tolerability of $k$ -ary $n$ -cube (denoted $Q_{n}<^>{k}$ ) networks. $Q_{n}<^>{k}$ is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the <italic>structure/substructure connectivity</italic> of $Q_{n}<^>{k}$ networks, for <italic>paths</italic> and <italic>cycles</italic>, two basic yet important network structures. Let $G$ be a connected graph and $T$ a connected subgraph of $G$ . The $T$ -<italic>structure connectivity</italic> $\kappa (G; T)$ of $G$ is the cardinality of a minimum set of subgraphs in $G$ , such that each subgraph is isomorphic to $T$ , and the sets removal disconnects $G$ . The $T$ -<italic>substructure connectivity</italic> $\kappa <^>{s}(G; T)$ of $G$ is the cardinality of a minimum set of subgraphs in $G$ , such that each subgraph is isomorphic to a connected subgraph of $T$ , and the sets removal disconnects $G$ . In this paper, we study $\kappa (Q_{n}<^>{k}; T)$ and $\kappa <^>{s}(Q_{n}<^>{k}; T)$ for $T=P_{i}$ , a path on $i$ nodes (resp. $T=C_{i}$ , a cycle on $i$ nodes). Lv <italic>et al.</italic> determined $\kappa (Q_{n}<^>{k}; T)$ and $\kappa <^>{s}(Q_{n}<^>{k}; T)$ for $T\in \{P_{1},P_{2},P_{3}\}$ . Our results generalize the preceding results by determining $\kappa (Q_{n}<^>{k}; P_{i})$ and $\kappa <^>{s}(Q_{n}<^>{k}; P_{i})$ . In addition, we have also established $\kappa (Q_{n}<^>{k}; C_{i})$ and $\kappa <^>{s}(Q_{n}<^>{k}; C_{i})$ .