The diagnosability of interconnection networks

Abstract

Diagnosability is a fundamental consideration when designing an interconnected network. The PMC and MM${}^{\ast }$ fault diagnosis models are the two most commonly used models. Both the $g$-good-neighbour diagnosability and $g$-extra diagnosability of an interconnection network have been two of the hot topics in the intersectional research areas of Graph theory and Computer Science, which become increasingly attractive for new solutions to real-world problems. However, there are still some problems in the transformation from the concepts of Computer Science to that of mathematics. In this paper, we systematically study such problems and give a strict proof from concepts to mathematical conclusions. In the terms of results, we not only give the relationship between $g$-good-neighbour diagnosabilities of the network under PMC model and MM${}^{\ast }$ model, but also between $g$-extra diagnosabilities of the network under PMC and MM${}^{\ast }$ models. To apply our results, we give an application on the enhanced hypercube in the end and derive a lemma explaining whether these are 3-cycles in enhanced hypercubes and how many common neighbours for two vertices of enhanced hypercubes under different values of $k$ in the meantime.