Necessary Condition for Path Partitioning Constraints

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Given a directed graph $\mathcal{G}$ , the K node-disjoint paths problem consists in finding a partition of $\mathcal{G}$ into K node-disjoint paths, such that each path ends up in a given subset of nodes in $\mathcal{G}$ . This article provides a necessary condition for the K node-disjoint paths problem which combines (1) the structure of the reduced graph associated with $\mathcal{G}$ , (2) the structure of each strongly connected component of $\mathcal{G}$ with respect to dominance relation between nodes, and (3) the way the nodes of two strongly connected components are inter-connected. This necessary condition is next used to deal with a path partitioning constraint.