On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks

We consider a directed acyclic network with unit-capacity edges and n source-terminal (s_i - t_i) pairs that wish to communicate at unit rate via network coding. The connectivity between the s_i-t_i pairs is quantified by means of a connectivity level vector, [k₁ k₂ ... k_n] such that there exist k_i edge-disjoint paths between s_i and t_i. In this work we attempt to classify networks based on the connectivity level. Specifically, for case of n = 3, and a given connectivity level vector [k₁ k₂ k₃], we aim to either present a constructive network coding scheme or an instance of a network that cannot support the desired rate. Certain generalizations for the case of higher n have also been found.