Conditional $(t,k)$ -Diagnosis in Regular and Irregular Graphs Under the Comparison Diagnosis Model

Abstract

Assume that there are at most $t$ faulty vertices. A system is conditionally ( $t,k$ )- diagnosable if at least $k$ faulty vertices (or all faulty vertices if fewer than $k$ faulty vertices remain) can be identified in each iteration under the assumption that every vertex is adjacent to at least one fault-free vertex. Let $\kappa _c(G)$ be the conditional vertex connectivity of $G$ , which measures the vertex connectivity of $G$ according to the assumption that every vertex is adjacent to at least one fault-free vertex. Let $\Delta (G)$ be the maximum degrees of the given graph $G$ . When a graph $G$ satisfies the condition that for any pair of vertices with distance two has at least two common neighbors in $G$ , we show the following two results: 1) An $r$ -regular network $G$ containing $N$ vertices is conditionally $\left(\frac{N+\sqrt{\frac{4\kappa (G)N}{(r+1)(r-1)}}-2}{r+1},\kappa _c(G)\right)$ -diagnosable, where $r \geq 3$ and $N \geq \frac{(r+1)(25r-9)}{4\kappa (G)}$ . 2) An irregular network $G$ containing $N$ vertices is conditionally $(\frac{N}{\Delta (G)+1}-1,\kappa _c(G))$ -diagnosable. By applying the above results to multiprocessor systems, we can measure conditional $(t,k)$ -diagnosabilities for augmented cubes, folded hypercubes, balanced hypercubes, and exchanged hypercubes.