Packing internally disjoint Steiner paths of data center networks

Let S ⊆ V ( G ) and π G ( S ) denote the maximum number t of edge-disjoint paths P 1 , P 2 , … , P t in a graph G such that V ( P i ) ∩ V ( P j ) = S for any i , j ∈ { 1 , 2 , … , t } and i ≠ j . If S = V ( G ) , then π G ( S ) is the maximum number of edge-disjoint spanning paths in G . It is proved [Graphs Combin, 37 (2021) 2521–2533] that deciding whether π G ( S ) ≥ r is NP-complete for a given S ⊆ V ( G ) . For an integer r with 2 ≤ r ≤ n , the r -path connectivity of a graph G is defined as π r ( G ) = min { π G ( S ) | S ⊆ V ( G ) and | S | = r } , which is a generalization of tree connectivity. In this paper, we study the 3-path connectivity of the k -dimensional data center network with n -port switches D k , n which has signification role in the cloud computing, and prove that π 3 ( D k , n ) = ⌊ 2 n + 3 k 4 ⌋ with k ≥ 0 and n ≥ 3 .