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An (n, N, d) -connector is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length d joining each input to its corresponding output. We consider the problem of construction of sparse (n, N, 2)-connectors (depth 2 connectors) when n N . The probabilistic argument in  shows the existence of (n, N, 2)connectors of size (number of edges) O(N ) if n ≤ N 1/2−ε , ε > 0. However, the knowndoi:10.1016/j.endm.2005.07.004 fatcat:ll7z2he47ngzrnys3seeuky65i