Structure Connectivity and Substructure Connectivity of $k$ -Ary $n$ -Cube Networks

We present new results on the fault tolerability of
$k$
-ary
$n$
-cube (denoted
$Q_{n}^{k}$
) networks.
$Q_{n}^{k}$
is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the
structure/substructure connectivity
of
$Q_{n}^{k}$
networks, for
paths
and
cycles
, two basic yet important network structures. Let
$G$
be a connected graph and
$T$
a connected subgraph of
$G$
. The
$T$
-
structure connectivity

$\kappa (G; T)$
of
$G$
is the cardinality of a minimum set of subgraphs in
$G$
, such that each subgraph is isomorphic to
$T$
, and the set’s removal disconnects
$G$
. The
$T$
-
substructure connectivity

$\kappa ^{s}(G; T)$
of
$G$
is the cardinality of a minimum set of subgraphs in
$G$
, such that each subgraph is isomorphic to a connected subgraph of
$T$
, and the set’s removal disconnects
$G$
. In this paper, we study
$\kappa (Q_{n}^{k}; T)$
and
$\kappa ^{s}(Q_{n}^{k}; T)$
for
$T=P_{i}$
, a path on
$i$
nodes (resp.
$T=C_{i}$
, a cycle on
$i$
nodes). Lv
et al.
determined
$\kappa (Q_{n}^{k}; T)$
and
$\kappa ^{s}(Q_{n}^{k}; T)$
for
$T\in \{P_{1},P_{2},P_{3}\}$
. Our results generalize the preceding results by determining
$\kappa (Q_{n}^{k}; P_{i})$
and
$\kappa ^{s}(Q_{n}^{k}; P_{i})$
. In addition, we have also established
$\kappa (Q_{n}^{k}; C_{i})$
and
$\kappa ^{s}(Q_{n}^{k}; C_{i})$
.