Reliability Analysis of the Generalized Exchanged Hypercube

Let G = (V(G), E(G)) be a non-complete graph, a subset T subset of V (G) is called a r-component cut of G, if G - T is disconnected and has at least r components. The cardinality of the minimum r-component cut is the r-component connectivity of G and is denoted by c kappa(r)(G). The r-component connectivity is a natural extension of the classical connectivity. As an application, the r-component connectivity can be used to evaluate the reliability and fault tolerance of an interconnection network structure based on a graph model. In a previous work, E. Cheng et al. obtained the r-component connectivity of the generalized exchanged hypercube GEH(s, t) for 1 <= r <= s and s >= 2. In this paper, we continue the work and determine that c kappa(s+2)(GEH(s, t)) = s(2) + 3s/2 - 1 for s >= 6. Moreover, we show that every optimal r-component cut of GEH(s, t) is trivial for 1 <= r <= s and s >= 2.