# Fault diagnosability of data center networks

## TL;DR

This paper investigates the diagnosability of data center networks, establishing a relationship between connectivity and diagnosability, and provides explicit diagnosability formulas for the networks $D_{k,n}$ under common models.

## Contribution

It solves an open problem by relating $R^g$-connectivity and $g$-good-neighbor diagnosability for regular graphs and derives explicit diagnosability formulas for $D_{k,n}$ networks.

## Key findings

- Established that $t_g(G)= ext{appa}^g(G)+g$ under certain conditions.
- Derived the $g$-good-neighbor diagnosability of $D_{k,n}$ as $(g+1)(k-1)+n+g$.
- Proved $D_{k,n}$ is tightly super $(n+k-1)$-connected and characterized the largest component after vertex removal.

## Abstract

The data center networks $D_{n,k}$, proposed in 2008, has many desirable features such as high network capacity. A kind of generalization of diagnosability for network $G$ is $g$-good-neighbor diagnosability which is denoted by $t_g(G)$. Let $\kappa^g(G)$ be the $R^g$-connectivity. Lin et. al. in [IEEE Trans. on Reliability, 65 (3) (2016) 1248--1262] and Xu et. al in [Theor. Comput. Sci. 659 (2017) 53--63] gave the same problem independently that: the relationship between the $R^g$-connectivity $\kappa^g(G)$ and $t_g(G)$ of a general graph $G$ need to be studied in the future. In this paper, this open problem is solved for general regular graphs. We firstly establish the relationship of $\kappa^g(G)$ and $t_g(G)$, and obtain that $t_g(G)=\kappa^g(G)+g$ under some conditions. Secondly, we obtain the $g$-good-neighbor diagnosability of $D_{k,n}$ which are $t_g(D_{k,n})=(g+1)(k-1)+n+g$ for $1\leq g\leq n-1$ under the PMC model and the MM model, respectively. Further more, we show that $D_{k,n}$ is tightly super $(n+k-1)$-connected for $n\geq 2$ and $k\geq 2$ and we also prove that the largest connected component of the survival graph contains almost all of the remaining vertices in $D_{k,n}$ when $2k+n-2$ vertices removed.