Directed graphs with unique paths of fixed length

Covering partially directed graphs with directed paths

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...

Cycle decompositions IV: complete directed graphs and fixed length directed cycles

We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m; where 2pmpn: r 2003 Elsevier Science (USA). All rights reserved.

Finding Disjoint Paths on Directed Acyclic Graphs

Given k + 1 pairs of vertices (s1, s2), (u1, v1), . . . , (uk, vk) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (ul, vl) with 1 ≤ l ≤ k, a tuple (s1, t1, s2, t2) with {t1, t2} = {ul, vl} in constant time such that there are two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such a tuple exists. Di...

Decomposing Semi-complete Multigraphs and Directed Graphs into Paths of Length Two

A P3-decomposition of a graph is a partition of the edges of the graph into paths of length two. We give a necessary and sufficient condition for a semi-complete multigraph, that is a multigraph with at least one edge between each pair of vertices, to have a P3-decomposition. We show that this condition can be tested in strongly polynomial-time, and that the same condition applies to a larger c...

Length-bounded disjoint paths in planar graphs