Fault Tolerance of k-Split Networks

Fault tolerance is especially important for interconnection networks, since the growing size of networks increases their vulnerability to component failures. A classical measure for the fault tolerance of a network in the case of vertex failures is its connectivity. The connectivity κ (G) of a connected graph G is the least positive integer κ such that there is S ⊂ V, |S| = k and G-S is disconnected or reduces to the trivial graph K
₁
. The edge connectivity λ(G) of G is similarly defined. It is well known that κ (G) ≤ λ (G) ≤ δ (G). A graph G is called maximally (edge) connected if κ (G) = δ (G)(λ (G) = δ (G)), max-κ (max-λ) for simply. A connected graph G = (V, E) is called supper-κ (resp. super-λ) if every minimum vertex cut (edge cut) of G is the set of neighbors of some vertex in G. We introduce a new kind of network called k-split network. A network based on a graph G = (X
₁
∪ X
₂
U ⋯∪ X
_k
, I, E) is called a k-split network, if its vertex set V can be partitioned into k + 1 stable sets I, X
₁
, X
₂
, ..., X
_k
such that X
₁
∪ X
₂
∪ ⋯ ∪ X
_k
induces a complete k-partite graph and an independent set I. In this note, we show that: G = (X
₁
∪ X
₂
⋯ ∪ X
_k
, I, E) is a non-complete connected k-split graph with k ≥ 3 and |X
₁
| ≤ |X
₂
| ≤ ⋯ ≤ |X
_k
|. Then we have (1) If |X
₁
∪ X
₂
⋯ ∪ X
_k-1
| ≥ δ(G), then κ (G) = δ (G). (2) If |X
₁
∪ X
₂
⋯ ∪ X
_k-1
|δ, then G is super-κ. (5) G is super-λ.