Disjoint Paths and Trees

In this chapter we concentrate on problems concerning (arc)-disjoint paths or trees (arborescences). We embark from the 2-path problem which concerns the existence of two disjoint paths with prescribed initial and terminal vertices. We give a proof by Fortune et al. showing that the 2-path problem is NP-complete. We proceed by studying the more general k-path problem for various classes of digraphs. We show that for acyclic digraphs, the k-path problem is polynomially solvable when k is not a part of the input. Then we describe several results on the k-path problem for generalizations of tournaments. Among other results, we show that the 2-path problem is polynomially solvable for digraphs that can be obtained from strong semi-complete digraphs by substituting arbitrary digraphs for each vertex of the semicomplete digraph. We briefly discuss the k-path problem for planar digraphs and indicate how to use the topological concept of planarity in proofs and algorithms for disjoint path problems in planar digraphs.