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, [k
₁
k
₂
... 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 [k
₁
k
₂
k
₃
], 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.