The $h$ -Extra Connectivity and Diagnosability of Locally Twisted Cubes

The connectivity and diagnosability of a system or a network are two important measures. In 1996, Fàbrega and Fiol proposed the
$h$
-extra connectivity of the network
$G=(V,E)$
, which is necessary for
$(h,m)$
-diagnosability of networks. In 2016, Zhang
et al.
proposed the
$(h,m)$
-diagnosability of
$G$
that requires every component of
$G-S$
has at least
$(h+1)$
nodes for
$S\subseteq V$
. The locally twisted cube
$LTQ_{n}$
is applied widely. There are many studies on
$LTQ_{n}$
. In this paper, we show that the
$h$
-extra connectivity of
$LTQ_{n}$
is
$n-\frac {1}{2}h(h-2n+3)$
for
$n\geq 5$
and
$0\leq h\leq n-3$
, and
$m$
of the
$(h,m)$
-diagnosability of
$LTQ_{n}$
is
$n-\frac {1}{2}h(h-2n+1)$
for
$n\geq 5$
,
$0\leq h\leq n-3$
in the PMC model and
$n\geq 7$
,
$0\leq h\leq n-3$
in the MM
$^{*}$
model, respectively.