Conditional (edge-)fault-tolerant strong Menger (edge) connectivity of folded hypercubes

Edge-fault-tolerant strong Menger edge connectivity on the class of hypercube-like networks

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 $\ell =4,5$ 
.

On Computing Component (Edge) Connectivities of Balanced Hypercubes

The generalized hypercube (GH) is one key interconnection network with excellent topological properties. It contains many other interconnection topologies, such as the hypercube network, the complete graph, the mesh network, and the $k$ k -ary $n$ n -cube network. It can also be used to construct some data center networks, such as HyperX, BCube, FBFLY, and SWCube. However, the construction cost of GH is high since it contains too many links. In this paper, we propose a novel low cost interconnection architecture called the exchanged generalized hypercube (EGH). We study the properties of EGH, such as the number of edges, the degree of vertices, connectivity, diameter, and diagnosability. Then, we give a routing algorithm to find the shortest path between any two distinct vertices of EGH. Furthermore, we design an algorithm to give disjoint paths between any two distinct vertices of EGH. In addition, we propose two local diagnosis algorithms: LDT $_{EGH}$ E G H and LDWB $_{EGH}$ E G H in EGH under PMC model and MM model, respectively. Simulation results demonstrate that even if the proportion of faulty vertices in EGH is up to 25 percent, the probability that these two diagnosis algorithms can successfully determine the status of vertices is more than 90 percent. As far as the number of edges is concerned, the analysis shows that the construction cost of EGH is much less than that of GH. We could regard this work as the basis for proposing future new high performance topologies.

Pingshan Li

Papers

Fault-tolerant strong Menger (edge) connectivity and 3-extra edge-connectivity of balanced hypercubes ☆

Conditional (edge-)fault-tolerant strong Menger (edge) connectivity of folded hypercubes