Conditional Diagnosability of Enhanced Hypercubes Under the PMC Model

In recent years, due to the continuous expansion of the scale of multi-processor systems, processor fault diagnosis has become increasingly important in measuring system reliability. Diagnosability of many well-known multiprocessor systems has been extensively studied. Conditional diagnosability is a new system diagnostic measure that inserts an additional condition that all neighbors (adjacent nodes) of any node in the system cannot all fail at the same time. In this report, we evaluate the conditional diagnosability for enhanced hypercubes under the PMC model. We first give several properties about hypercubes and enhanced hypercubes, and then based on these properties, the conditional diagnosability of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula>-enhanced hypercube <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{n,k}$ </tex-math></inline-formula> is proved to be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4n-7$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\ge 5$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=3$ </tex-math></inline-formula>, and to be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4n-3$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\ge 6$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5\le k\le n$ </tex-math></inline-formula>.