Relating Diagnosability, Strong Diagnosability and Conditional Diagnosability of Strong Networks

An interconnection network’s diagnosability is an important measure of its self-diagnostic capability. Based on the classical notion of diagnosability, strong diagnosability and conditional diagnosability were proposed later to better reflect the networks’ self-diagnostic capability under more realistic assumptions. In this paper, we study a class of interconnection networks called strong networks, which are
$n$
-regular,
$(n - 1)$
-connected, and with
$cn$
-number no more than
$n - 3$
. We build a relationship among the three diagnosability measures for strong networks. Under both PMC and
${\rm MM}^{\ast}$
models, given a strong network
$G$
with diagnosability
$t$
, we prove that
$G$
is strongly
$t$
-diagnosable if and only if
$G$
’s conditional diagnosability is greater than
$t$
. A simple check can show that almost all well-known regular interconnection networks are strong networks. The significance of this paper’s result is that it reveals an important relationship between strong and conditional diagnosabilities, and the proof of strong diagnosability for many interconnection networks under
${\rm MM}^{\ast}$
or PMC model is not necessary if their conditional diagnosability can be shown to be strictly larger than their diagnosability.