A Pessimistic Fault Diagnosability of Large-Scale Connected Networks via Extra Connectivity

Limei Lin, Yanze Huang, Li Xu, Sun-Yuan Hsieh

Abstract: The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t/kt/k-diagnosability</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hh-extra connectivity</i> are regarded as two important indicators to improve the network reliability. The t/k-diagnosis strategy can significantly improve the self-diagnosing capability of a network at the expense of no more than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> fault-free nodes being mistakenly diagnosed as faulty. The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -extra connectivity can tremendously improve the real fault tolerability of a network by insuring that each remaining component has no fewer than h+1 nodes. However, there is few result on the inherent relationship between these two indicators. In this article, we investigate the reason that caused the serious flawed results in (Liu, 2020), and we propose a diagnosis algorithm to establish the t/k-diagnosability for a large-scale connected network <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> under the PMC model by considering its h-extra connectivity. Let κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sub> (G) be the h-extra connectivity of G. Then, we can deduce that G is κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sub> (G)/h-diagnosable under the PMC model with some basic conditions. All κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sub> (G)faulty nodes can be correctly diagnosed in the large-scale connected network G and at most h fault-free nodes would be misdiagnosed as faulty. The complete fault tolerant method adopts combinatorial properties and linearly many fault analysis to conquer the core of our proofs. We will apply the newly found relationship to directly obtain the κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sub> (G)/h-diagnosability of a series of well known networks, including hypercubes, folded hypercubes, balanced hypercubes, dual-cubes, BC graphs, star graphs, Cayley graphs generated by transposition trees, bubble-sort star graphs, alternating group graphs, split-star networks, k-ary n-cubes and (n,k)-star graphs.