We consider the multiple-unicast problem with three source–terminal pairs over directed acyclic networks with unit-capacity edges. The three – pairs wish to communicate at unitrate via network coding. The connectivity between the – pairs is quantified by means of a connectivity-level vector, such that there exist edge-disjoint paths between and . In this paper, we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if , for all . In this paper, we consider connectivity-level vectors such that . We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unitrate 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.