We consider graphs which contain both directed and undirected edges (partially directed graphs). We show that the problem of covering the edges of such graphs with a minimum number of edge-disjoint directed paths respecting the orientations of the directed edges is polynomially solvable.We exhibit a good characterization for this problem in the form of a min–max theorem.We introduce a more gene...

We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [ 31 ] problems undirected graphs. The main insight is that approximately balanced cost functions can be solved similarly to case. finds reduced function O(m √ log ) preprocessing step. Using the...