Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs

TL;DR

This paper explores the relationship between conditional diagnosability and 2-extra connectivity in symmetric graphs, establishing their equality under certain conditions and applying findings to various well-known network classes.

Contribution

It proves that for regular graphs satisfying specific conditions, the conditional diagnosability equals the 2-extra connectivity, and determines these parameters for multiple classes of vertex-transitive graphs.

Findings

Proves $t_c(G)=2(G)$ for certain regular graphs.

Determines conditional diagnosability and 2-extra connectivity for various well-known networks.

Provides new insights and results for the fault-tolerance parameters of symmetric graphs.

Abstract

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability $t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally $t$-diagnosable under the comparison model, while the 2-extra connectivity $\kappa_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$ vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answer this problem by proving $t_c(G)=\kappa_2(G)$ for a regular graph $G$ with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, $(n,k)$-star graphs, alternating group networks, $(n,k)$-arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, $k$-ary $n$-cube networks and dual-cubes. Furthermore, many known results about these networks are obtained directly.