The Extra Connectivity, Extra Conditional Diagnosability, and <inline-formula> <tex-math notation="LaTeX"> $t/m$</tex-math> </inline-formula>-Diagnosability of Arrangement Graphs

Abstract

Extra connectivity is an important indicator of the robustness of a multiprocessor system in presence of failing processors. The $g$ -extra conditional diagnosability and the $t/m$ -diagnosability are two important diagnostic strategies at system-level that can significantly enhance the system's self-diagnosing capability. The $g$ -extra conditional diagnosability is defined under the assumption that every component of the system removing a set of faulty vertices has more than $g$ vertices. The $t/m$ -diagnosis strategy can detect up to $t$ faulty processors which might include at most $m$ misdiagnosed processors, where $m$ is typically a small integer number. In this paper, we analyze the combinatorial properties and fault tolerant ability for an $(n,k)$ -arrangement graph, denoted by $A_{n,k}$ , a well-known interconnection network proposed for multiprocessor systems. We first establish that the $A_{n,k}$ 's one-extra connectivity is $(2k-1) (n-k)-1$ ( $k\geq 3$ , $n\geq k+2$ ), two-extra connectivity is $(3k-2)(n-k)-3$ ( $k\geq 4$ , $n\geq k+2$ ), and three-extra connectivity is $(4k-4)(n-k)-4$ ( $k\geq 4$ , $n\geq k+2$ or $k\geq 3$ , $n\geq k+3$ ), respectively. And then, we address the $g$ -extra conditional diagnosability of $A_{n,k}$ under the PMC model for $1\leq g \leq 3$ . Finally, we determine that the $(n,k)$ -arrangement graph $A_{n,k}$ is $[(2k-1)(n-k)-1]/1$ -diagnosable ( $k\geq 4$ , $n\geq k+2$ ), $[(3k-2)(n-k)-3]/2$ -diagnosable ( $k\geq 4$ , $n\geq k+2$ ), and $[(4k-4)(n-k)-4]/3$ -diagnosable ( $k\geq 4$ , $n\geq k+3$ ) under the PMC model, respectively.