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The $g$-good-neighbor diagnosability of product networks under the PMC model

Zhao Wang, Yaping Mao, Sun-Yuan Hsieh, Ralf Klasing

TL;DR

This paper analyzes the $g$-good-neighbor diagnosability of graphs under the PMC model, introducing the gc-number $c^{g}(G)$ and establishing two critical existence criteria for $t^{g}(G)$. It proves sharp bounds linking $t^{g}(G)$ to $c^{g}(G)$ and $ abla^{g}(G)$, and extends the theory to Cartesian product networks, providing comprehensive, sharp upper bounds for $t^{g}(G\Box H)$ in multiple cases. The authors apply the framework to 2D grid, torus, and generalized hypercube networks, deriving exact values or tight bounds for $c^{g}$ and $t^{g}$ and demonstrating the approach’s practicality for key interconnection networks. Overall, the work yields computable diagnosability metrics for general and product networks, with potential impact on design and testing of fault-tolerant multiprocessor systems.

Abstract

The concept of neighbor connectivity originated from the assessment of the subversion of espionage networks caused by underground resistance movements, and it has now been applied to measure the disruption of networks caused by cascading failures through neighbors. In this paper, we give two necessary and sufficient conditions of the existance of $g$-good-neighbor diagnosability. We introduce a new concept called $g$-good neighbor cut-component number (gc number for short), which has close relation with $g$-good-neighbor diagnosability. Sharp lower and upper bounds of the gc number of general graphs in terms of the $g$-good neighbor connectivity is given, which provides a formula to compute the $g$-good-neighbor diagnosability for general graphs (therefore for Cartesian product graphs). As their applications, we get the exact values or bounds for the gc numbers and $g$-good-neighbor diagnosability of grid, torus networks and generalized cubes.

The $g$-good-neighbor diagnosability of product networks under the PMC model

TL;DR

This paper analyzes the -good-neighbor diagnosability of graphs under the PMC model, introducing the gc-number and establishing two critical existence criteria for . It proves sharp bounds linking to and , and extends the theory to Cartesian product networks, providing comprehensive, sharp upper bounds for in multiple cases. The authors apply the framework to 2D grid, torus, and generalized hypercube networks, deriving exact values or tight bounds for and and demonstrating the approach’s practicality for key interconnection networks. Overall, the work yields computable diagnosability metrics for general and product networks, with potential impact on design and testing of fault-tolerant multiprocessor systems.

Abstract

The concept of neighbor connectivity originated from the assessment of the subversion of espionage networks caused by underground resistance movements, and it has now been applied to measure the disruption of networks caused by cascading failures through neighbors. In this paper, we give two necessary and sufficient conditions of the existance of -good-neighbor diagnosability. We introduce a new concept called -good neighbor cut-component number (gc number for short), which has close relation with -good-neighbor diagnosability. Sharp lower and upper bounds of the gc number of general graphs in terms of the -good neighbor connectivity is given, which provides a formula to compute the -good-neighbor diagnosability for general graphs (therefore for Cartesian product graphs). As their applications, we get the exact values or bounds for the gc numbers and -good-neighbor diagnosability of grid, torus networks and generalized cubes.

Paper Structure

This paper contains 15 sections, 23 theorems, 42 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diagnosis models
  4. $g$-good-neighbor connectivity and diagnosability
  5. Cartesian product networks
  6. Necessary and sufficient conditions
  7. $g$-good neighbour cut-component numbers
  8. Results for $g$-good-neighbor diagnosability
  9. Results for $g$-good-neighbor diagnosability of product networks
  10. Applications
  11. 2-dimensional grid networks
  12. 2-dimensional torus networks
  13. 2-dimensional generalized hypercube
  14. Concluding Remark
  15. Acknowledgments

Key Result

Theorem 1.1

Let $G$ be a graph. For each paired distinct sets $F_1, F_2 \subset V, F_1$ and $F_2$ are distinguishable under the PMC model if and only if there exist $u \in F_1 \triangle F_2$ and $v \in \overline{F_1 \cup F_2}$ such that $u v \in E$; see Fig. Figure1.

Figures (9)

  • Figure 1: Graphs for Theorem \ref{['th-DM84']}.
  • Figure 2: Graphs for Remark \ref{['rem1-1']}
  • Figure 3: Two subsets $X_1$ and $X_2$, where "$\times$" denote the Cartesian product $\Box$.
  • Figure 4: The edge $(u,v)(u',v)$ in $G\Box H$, where "$\times$" denote the Cartesian product $\Box$.
  • Figure 5: Graphs for Case \ref{['case2']}, where "$\times$" denote the Cartesian product $\Box$.
  • ...and 4 more figures

Theorems & Definitions (56)

  • Theorem 1.1: Dahbura and Masson, 1984, DM84
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.1
  • Definition 1.5
  • Definition 1.6
  • Remark 2.1
  • Theorem 2.1
  • ...and 46 more