  We consider the multiple unicast problem with three source-terminal pairs
over directed acyclic networks with unit-capacity edges. The three $s_i-t_i$
pairs 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, $[k_1 k_2 k_3]$ 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. It can be observed that unit-rate transmission can be
supported by routing if $k_i \geq 3$, for all $i = 1, \dots, 3$. In this work,
we consider, connectivity level vectors such that $\min_{i = 1, \dots, 3} k_i <
3$. We present either a constructive linear network coding scheme or an
instance of a network that cannot support the desired unit-rate requirement,
for all such connectivity level vectors except the vector $[1~2~4]$ (and its
permutations). The benefits of our schemes extend to networks with higher and
potentially different edge capacities. Specifically, our experimental results
indicate that for networks where the different source-terminal paths have a
significant overlap, our constructive unit-rate schemes can be packed along
with routing to provide higher throughput as compared to a pure routing
approach.
