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Structure and substructure connectivity of divide-and-swap cube
Theoretical Computer Science  (IF0.827),  Pub Date : 2021-06-10, DOI: 10.1016/j.tcs.2021.05.033
Qianru Zhou, Shuming Zhou, Jiafei Liu, Xiaoqing Liu

High fault tolerance and reliability of multiprocessor systems, modeled by interconnection network, are of great significance to assess the flexibility and effectiveness of the systems. Connectivity is an important metric to evaluate the fault tolerance and reliability of interconnection networks. As classical connectivity is not suitable for such large scale systems, a novel and generalized connectivity, structure connectivity and substructure connectivity, has been proposed to measure the robustness of networks and has witnessed rich achievements. The divide-and-swap cube $DS{C}_{n}$ is an interesting variant of hypercube that has nice hierarchical properties. In this paper, we mainly investigate $\mathcal{H}$-structure-connectivity, denoted by $\kappa \left(DS{C}_{n};\mathcal{H}\right)$, and $\mathcal{H}$-substructure-connectivity, denoted by ${\kappa }^{s}\left(DS{C}_{n};\mathcal{H}\right)$, for $\mathcal{H}\in \left\{{K}_{1},{K}_{1,1},{K}_{1,m}\left(2\le m\le d+1\right),{C}_{4}\right\}$, respectively. In detail, we show that $\kappa \left(DS{C}_{n};{K}_{1}\right)={\kappa }^{s}\left(DS{C}_{n};{K}_{1}\right)=d+1$ for $n\ge 2$, $\kappa \left(DS{C}_{n};{K}_{1,1}\right)={\kappa }^{s}\left(DS{C}_{n};{K}_{1,1}\right)=d+1$ for $n\ge 8$, $\kappa \left(DS{C}_{n};{K}_{1,m}\right)={\kappa }^{s}\left(DS{C}_{n};{K}_{1,m}\right)=⌊\frac{d}{2}⌋+1$ with $2\le m\le d+1$ for $n\ge 4$, $\kappa \left(DS{C}_{n};{C}_{4}\right)=3+2\left(d-2\right)$ for $4\le n\le 8$, $⌊\frac{d}{2}⌋+1\le \kappa \left(DS{C}_{n};{C}_{4}\right)\le d+1$ for $n\ge 16$ and ${\kappa }^{s}\left(DS{C}_{n};{C}_{4}\right)=⌊\frac{d}{2}⌋+1$ for $n\ge 4$. Finally, we compare and analyze the ratios of structure (resp. substructure) connectivity to vertex degree of divide-and-swap cube with that of several well-known variants of hypercube.