The k-restricted edge-connectivity of the data center network DCell

For any integers m >= 0 and n >= 2, we use D-m,D-n to denote the m-dimensional DCell with nport switches. For a simple graph G = (V, E), an edge subset S subset of E(G) is said to be a lambda(k)-cut of G, if G - S is disconnected and each component of G - S has at least k vertices. The k-restricted edge-connectivity of G, denoted by lambda(k)(G), is the minimum cardinality of all lambda(k) cuts of G. We prove that lambda(k) (D-m,D-n) = k (m + n - k) for m >= n >= 2 and 2 <= k <= n, lambda(k) (D-m,D-n) = mn for n > m >= 2 and m <= k <= n and lambda(k)(D-m,D-n) = k(m + n - k) for n > m >= 2 and 2 <= k < m. We also characterize the structure of the minimum lambda(k) -cuts of D-m,D-n in terms of the above conclusions. (C) 2020 Published by Elsevier Inc.