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When there exists only a single multicast session in a directed acyclic/cyclic network, the existence of a network coding solution is characterized by the classic min-cut/max-flow theorem. For the case of more than one coexisting sessions, network coding also demonstrates throughput improvement over noncoded solutions. This paper proposes pairwise intersession network coding, which allows for arbitrary directed networks but restricts the coding operations to being between two symbols (for acyclic networks) or between two strings of symbols (for cyclic networks). A graph-theoretic characterization of pairwise intersession network coding is proven based on paths with controlled edge-overlap. This new characterization generalizes the edge-disjoint path characterization of noncoded network communication and includes the well-studied butterfly graph as a special case. Based on this new characterization, various aspects of pairwise intersession network coding are studied, including the sufficiency of linear codes, the complexity of identifying coding opportunities, its topological analysis, and bandwidth- and coding-efficiency.