The <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula>-Extra Connectivity and Diagnosability of Locally Twisted Cubes

Abstract

The connectivity and diagnosability of a system or a network are two important measures. In 1996, Fàbrega and Fiol proposed the $h$ -extra connectivity of the network $G=(V,E)$ , which is necessary for $(h,m)$ -diagnosability of networks. In 2016, Zhang et al. proposed the $(h,m)$ -diagnosability of $G$ that requires every component of $G-S$ has at least $(h+1)$ nodes for $S\subseteq V$ . The locally twisted cube $LTQ_{n}$ is applied widely. There are many studies on $LTQ_{n}$ . In this paper, we show that the $h$ -extra connectivity of $LTQ_{n}$ is $n-\frac {1}{2}h(h-2n+3)$ for $n\geq 5$ and $0\leq h\leq n-3$ , and $m$ of the $(h,m)$ -diagnosability of $LTQ_{n}$ is $n-\frac {1}{2}h(h-2n+1)$ for $n\geq 5$ , $0\leq h\leq n-3$ in the PMC model and $n\geq 7$ , $0\leq h\leq n-3$ in the MM $^{*}$ model, respectively.

Keywords

• Network
• connectivity
• diagnosability
• locally twisted cube