On Computing Component (Edge) Connectivities of Balanced Hypercubes
The Computer Journal(2020)
摘要
For an integer
$\ell \geqslant 2$
, the
$\ell $
-component connectivity (resp.
$\ell $
-component edge connectivity) of a graph
$G$
, denoted by
$\kappa _{\ell }(G)$
(resp.
$\lambda _{\ell }(G)$
), is the minimum number of vertices (resp. edges) whose removal from
$G$
results in a disconnected graph with at least
$\ell $
components. The two parameters naturally generalize the classical connectivity and edge connectivity of graphs defined in term of the minimum vertex-cut and the minimum edge-cut, respectively. The two kinds of connectivities can help us to measure the robustness of the graph corresponding to a network. In this paper, by exploring algebraic and combinatorial properties of
$n$
-dimensional balanced hypercubes
$BH_n$
, we obtain the
$\ell $
-component (edge) connectivity
$\kappa _{\ell }(BH_n)$
(
$\lambda _{\ell }(BH_n)$
). For
$\ell $
-component connectivity, we prove that
$\kappa _2(BH_n)=\kappa _3(BH_n)=2n$
for
$n\geq 2$
,
$\kappa _4(BH_n)=\kappa _5(BH_n)=4n-2$
for
$n\geq 4$
,
$\kappa _6(BH_n)=\kappa _7(BH_n)=6n-6$
for
$n\geq 5$
. For
$\ell $
-component edge connectivity, we prove that
$\lambda _3(BH_n)=4n-1$
,
$\lambda _4(BH_n)=6n-2$
for
$n\geq 2$
and
$\lambda _5(BH_n)=8n-4$
for
$n\geq 3$
. Moreover, we also prove
$\lambda _\ell (BH_n)\leq 2n(\ell -1)-2\ell +6$
for
$4\leq \ell \leq 2n+3$
and the upper bound of
$\lambda _\ell (BH_n)$
we obtained is tight for
$\ell =4,5$
.
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关键词
interconnection networks,generalized connectivity,component connectivity,component edge connectivity,balanced hypercubes
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