Reliability of augmented k-ary n-cubes under the extra connectivity condition

Abstract

Fault tolerance is critical to the reliability analysis of interconnection networks because the vulnerability of components increases with the growth of network scale. Extra connectivity and extra diagnosability are two decisive indicators to measure network fault tolerance and diagnostic capability. Recently, the extra fault tolerance of many triangle-free networks has been widely studied. However, many social networks, ad hoc networks, and complex networks are designed with girth 3 as the basic topology. At present, the extra fault tolerance analysis of such networks has not been studied. Therefore, this paper mainly discusses the extra fault tolerance of the augmented k-ary n-cube $AQ_{n, k}$ with girth 3, including the g-extra connectivity and the g-extra diagnosability. In detail, the g-extra connectivity of $AQ_{n, k}$ is $4n(1+g)-\lfloor \frac{5(1+g)^{2}}{2}\rfloor$ ($n\ge 4$, $k\ge 4$, and $0\le g\le n-2$), and the g-extra diagnosability of $AQ_{n, k}$ is $4n(1+g)-\lfloor \frac{5(1+g)^{2}}{2}\rfloor +g$ under the PMC model ($n\ge 4$, $k\ge 4$, and $0\le g\le n-2$) and the MM* model ($n\ge 7$, $k\ge 4$, and $1\le g\le \frac{n-5}{2}$). In addition, we explore the diagnosis algorithm of $AQ_{n, k}$ based on extra faults.