Reliability of [formula omitted]-ary [formula omitted]-dimensional hypercubes under embedded restriction

Abstract

The topological structure of a recursive interconnection network can be modeled by an n-dimensional graph Gn. A faulty vertex set (resp. edge set) of Gn is called a t-embedded vertex (resp. edge) cut if the remaining network is disconnected and each vertex in the residual network is in a copy of fault-free t-dimensional graph Gt. The t-embedded vertex connectivity ζt(Gn) (resp. the t-embedded edge connectivity ηt(Gn)) is defined as the minimum cardinality over all t-embedded vertex (resp. edge) cuts of Gn. The m-ary n-dimensional hypercube G(n,m) has an elegant recursive structure and is also a generalized variant of the classical hypercube. This paper studies the reliability of the G(n,m) and generalizes some known results in ternary n-cubes. Specifically, we prove that ζt(G(n,m))=(m−1)(n−t)mt for t≤n−2 and ηt(G(n,m))=(m−1)(n−t)mt for t≤n−1.