Hyper star structure fault tolerance of half hypercube

The reliability of interconnection networks ordinarily is measured by all kinds of connectivities. The qualitative and quantitative analysis relies on the assumptions consistent with the actual situation. Structure connectivity and substructure connectivity are key indicators to assess network reliability and fault tolerance. In this paper, we first determine the \(K_{1,r}\) $K_{1,r}$ -structure connectivity and \(K_{1,r}\) $K_{1,r}$ -substructure connectivity of half hypercube \(HH_n\) $H{H}_n$ for \(1\le r\le \lfloor \frac{n}{2}\rfloor -2\) $1\le r\le \lfloor \frac{n}{2}\rfloor -2$ . Then, we characterize the minimum \(K_{1,r}\) $K_{1,r}$ -cuts and the minimum sub- \(K_{1,r}\) $K_{1,r}$ -cuts of \(HH_n\) $H{H}_n$ . Specifically, when n is even, half hypercube \(HH_n\) $H{H}_n$ is isomorphic to hierarchical cubic network \(HCN_{\frac{n}{2}}\) $HC{N}_{\frac{n}{2}}$ , then our results on \(HH_n\) $H{H}_n$ can be directly applied to \(HCN_{\frac{n}{2}}\) $HC{N}_{\frac{n}{2}}$ .