# On $g$-good-neighbor conditional diagnosability of $(n, k)$-star networks

## TL;DR

This paper extends the understanding of $g$-good-neighbor conditional diagnosability for $(n,k)$-star networks under the PMC and MM* models, covering all remaining cases and generalizing previous results.

## Contribution

It determines the $g$-good-neighbor conditional diagnosability for all remaining cases of $(n,k)$-star networks, broadening prior partial results under two fault diagnosis models.

## Key findings

- Complete characterization of $t_g(S_{n,k})$ for all $1 \leq g \leq n-1$ and $1 \leq k \leq n-1$.
- Generalization of diagnosability results to the star graph case.
- Enhanced understanding of fault diagnosis capabilities in complex network topologies.

## Abstract

The $g$-good-neighbor conditional diagnosability is a new measure for fault diagnosis of systems. Xu et al. [Theor. Comput. Sci. 659 (2017) 53--63] determined the $g$-good-neighbor conditional diagnosability of $(n, k)$-star networks $S_{n, k}$ (i.e., $t_g(S_{n, k})$) with $1\leq k\leq n-1$ for $1\leq g\leq n-k$ under the PMC model and the MM$^*$ model. In this paper, we determine $t_g(S_{n, k})$ for all the remaining cases with $1\leq k\leq n-1$ for $1\leq g\leq n-1$ under the two models, from which we can obtain the $g$-good-neighbor conditional diagnosability of the star graph obtained by Li et al. [to appear in Theor. Comput. Sci.] for $1\leq g\leq n-2$.