6 research outputs found
Structure and substructure connectivity of balanced hypercubes
The connectivity of a network directly signifies its reliability and
fault-tolerance. Structure and substructure connectivity are two novel
generalizations of the connectivity. Let be a subgraph of a connected graph
. The structure connectivity (resp. substructure connectivity) of ,
denoted by (resp. ), is defined to be the minimum
cardinality of a set of connected subgraphs in , if exists, whose
removal disconnects and each element of is isomorphic to (resp. a
subgraph of ). In this paper, we shall establish both and
of the balanced hypercube for
.Comment: arXiv admin note: text overlap with arXiv:1805.0846
The restricted -connectivity of balanced hypercubes
The restricted -connectivity of a graph , denoted by , is
defined as the minimum cardinality of a set of vertices in , if exists,
whose removal disconnects and the minimum degree of each component of
is at least . In this paper, we study the restricted -connectivity of the
balanced hypercube and determine that
for . We also obtain a sharp upper
bound of and of -dimension balanced
hypercube for (). In particular, we show that
Fault-tolerant analysis of augmented cubes
The augmented cube , proposed by Choudum and Sunitha [S. A. Choudum, V.
Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a -regular
-connected graph . This paper determines that the 2-extra
connectivity of is for and the 2-extra
edge-connectivity is for . That is, for
(respectively, ), at least vertices (respectively,
edges) of have to be removed to get a disconnected graph that contains
no isolated vertices and isolated edges. When the augmented cube is used to
model the topological structure of a large-scale parallel processing system,
these results can provide more accurate measurements for reliability and fault
tolerance of the system
Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs
The conditional diagnosability and the 2-extra connectivity are two important
parameters to measure ability of diagnosing faulty processors and
fault-tolerance in a multiprocessor system. The conditional diagnosability
of is the maximum number for which is conditionally
-diagnosable under the comparison model, while the 2-extra connectivity
of a graph is the minimum number for which there is a
vertex-cut with such that every component of has at least
vertices. A quite natural problem is what is the relationship between the
maximum and the minimum problem? This paper partially answer this problem by
proving for a regular graph with some acceptable
conditions. As applications, the conditional diagnosability and the 2-extra
connectivity are determined for some well-known classes of vertex-transitive
graphs, including, star graphs, -star graphs, alternating group
networks, -arrangement graphs, alternating group graphs, Cayley graphs
obtained from transposition generating trees, bubble-sort graphs, -ary
-cube networks and dual-cubes. Furthermore, many known results about these
networks are obtained directly