The 3∗-connected property of pyramid networks

A k-container C(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) of G is a k∗-container if it contains all the vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices in G. Let κ(G) be the connectivity of G. A graph G is superconnected if G is i∗-connected for all 1 ≤ i ≤ κ(G). The pyramid network is one of the important networks applied in parallel and distributed computer systems. The connectivity of a pyramid network is three. In this paper, we prove that the pyramid network PM[n] is 3∗-connected and superconnected for n ≥ 1.