Reliability of m-ary n-dimensional hypercubes under embedded restriction

The topological structure of a recursive interconnection network can be modeled by an n-dimensional graph G(n). A faulty vertex set (resp. edge set) of G(n) 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 G(t). The t-embedded vertex connectivity zeta(t)(G(n)) (resp. the t-embedded edge connectivity eta(t)(G(n))) is defined as the minimum cardinality over all t-embedded vertex (resp. edge) cuts of G(n). 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 zeta(t)(G(n, m)) = (m - 1)(n - t)m(t) for t <= n - 2 and eta(t)(G(n, m)) = (m - 1)(n - t)m(t )for t <= n -1. (c) 2024 Elsevier B.V. All rights reserved.