3 research outputs found
Star Structure Connectivity of Folded hypercubes and Augmented cubes
The connectivity is an important parameter to evaluate the robustness of a
network. As a generalization, structure connectivity and substructure
connectivity of graphs were proposed. For connected graphs and , the
-structure connectivity (resp. -substructure connectivity
) of is the minimum cardinality of a set of subgraphs
of that each is isomorphic to (resp. to a connected subgraph of ) so
that is disconnected or the singleton. As popular variants of hypercubes,
the -dimensional folded hypercubes and augmented cubes are
attractive interconnected network prototypes for multiple processor systems. In
this paper, we obtain that
for , , and
for
The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes
As a generalization of vertex connectivity, for connected graphs and ,
the -structure connectivity (resp. -substructure
connectivity ) of is the minimum cardinality of a set of
subgraphs of that each is isomorphic to (resp. to a connected
subgraph of ) so that is disconnected. For -dimensional hypercube
, Lin et al. [6] showed
and
for
and . Sabir et al. [11] obtained that
for
, and for -dimensional folded hypercube ,
,
with and . They proposed an open problem of
determining -structure connectivity of and for general
. In this paper, we obtain that for each integer ,
and
for all integers larger than in quare scale. For , we
separately confirm the above result holds for in the remaining cases
Fault tolerance of hypercubes and folded hypercubes
NSFC [11301440, 11171279]; Xiamen University of Technology [YKJ12030R]; Foundation to the Educational Committee of Fujian [JA13240, JA13025, JA13034]; Natural Science Foundation of Fujian Province [2013J05006]; Natural Sciences Foundation of Guangxi Province [2012GXNSFBA053005, 2011GXNSFA018144]Let G = (V, E) be a connected graph. The conditional edge connectivity is the cardinality of the minimum edge cuts, if any, whose deletion disconnects and each component of has . We assume that is an edge set, is called edge extra-cut, if is not connected and each component of has more than vertices. The edge extraconnectivity is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes