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/mdiagnosability are two important diagnostic strategies at systemlevel 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/mdiagnosis 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) 3 (k ≥ 4, n ≥ k + 2), and three-extra connectivity is (4k − 4)(n − k) − 4 ( k ≥ 4, n ≥ k + 2 or k ≥ 3, n ≥ k + 3), respectively. And then, we address the g-extra conditional diagnosability of A n,k under the PMC model for 1 ≤ g ≤ 3. Finally, we determine that the (n, k)-arrangement graph A n,k is [(2k − 1)(n − k) − 1]/1-diagnosable (k ≥ 4, n ≥ k + 2), [(3k − 2)(n − k) − 3]/2-diagnosable (k ≥ 4, n ≥ k + 2), and [(4k − 4)(n − k) − 4]/3-diagnosable (k ≥ 4, n ≥ k + 3) under the PMC model, respectively.

The 2-good-neighbor connectivity and 2-good-neighbor diagnosability of bubble-sort star graph networks

“…They consider the situation that any fault set cannot contain all the neighbor vertices of any vertex in a system. The restricted diagnosability of the system has received much attention [5,6,20,24,25,38]. In 2012, Peng et al [27] proposed a measure for fault diagnosis of the system, namely, the g-good-neighbor diagnosability (which is also called the ggood-neighbor conditional diagnosability), which requires that every fault-free node has at least g fault-free neighbors.…”

Minimum Neighborhood of Alternating Group Graphs

“…In 2016, Zhao et al [39] proposed that the size of neighbors of an independent set D of vertices with |D| = q in Q n is ≥ −q 2 /2 + (2n − 5/2)q − n 2 +2n+1 for n+1 ≤ q ≤ 2n−4 and applied it to obtain the component connectivity. In 2016, Xu et al [32] established the lower bound of size of neighbors of a subset D with 3 ≤ |D| ≤ 5 in (n, k)-arrangement graph, which can be applied to t/k-diagnosability and extra conditional diagnosability. Lin et al [21] proposed the size of neighbors of a 2-path or a 3-cycle in general regular graphs, which can be applied to build the relationship between conditional diagnosability and extra connectivity under the comparison model.…”

Structure and Substructure Connectivity of Balanced Hypercubes

“…It is known that there exists no polynomial time algorithm to compute the g-connectivity and h-connectivity of a general graph [2,6]. The h-connectivity [4,14,23,24,26,35] and g-connectivity [3,10,15,28,34,36] of some famous networks are investigated in the literature.…”