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Super connected direct product of graphs and cycles

The topology of an interconnection network can be modeled by a graph $G=(V(G),E(G))$. The connectivity of graph $G$ is a parameter to measure the reliability of corresponding network. Direct product is one important graph product. This paper mainly focuses on the super connectedness of direct product of graphs and cycles. The connectivity of $G$, denoted by $κ(G)$, is the size of a minimum vertex set $S\subseteq V(G)$ such that $G-S$ is not connected or has only one vertex. The graph $G$ is said to be super connected, simply super-$κ$, if every minimum vertex cut is the neighborhood of a vertex with minimum degree. The direct product of two graphs $G$ and $H$, denoted by $G\times H$, is the graph with vertex set $V(G \times H) = V (G)\times V (H)$ and edge set $E(G \times H) = \{(u_{1} ,v_{1} )(u_{2} ,v_{2} )|\ u_{1}u_{2} \in E(G), v_{1}v_{2} \in E(H)\}$. In this paper, we give some sufficient conditions for direct product $G\times C_{n}$ to be super connected, where $C_{n}$ is the cycle on $n$ vertices. Furthermore, those sufficient conditions are best possible.