Arc-Disjoint Paths in Decomposable Digraphs

Disjoint paths in arborescences

An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have λ arc-disjoint arborescences rooted at r. A similar theorem of Menger guarantees λ strongly arc disjoint rv-paths for every vertex v, where “strongly” means no two paths contain a pair of symmetric arcs. We prove that if a directed graph D contains two arc-disjoin...

Arc-Disjoint Paths in Expander Digraphs

Arc-Disjoint Paths and Trees in 2-Regular Digraphs

An out-(in-)branching B s (B − s ) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x 6= s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: • Given a 2-regular digraph D, decide if it contains two arc-disjoint branchings B u , B − v . • Given a 2-regular digraph D,...

Complete classification of tournaments having a disjoint union of directed paths as a minimum feedback arc set

A feedback arc set of a digraph is a set of arcs whose reversal makes the resulting digraph acyclic. Given a tournament with a disjoint union of directed paths as a feedback arc set, we present necessary and sufficient conditions for this feedback arc set to have minimum size. We will present a construction for tournaments where the difference between the size of a minimum feedback arc set and ...

Complexity of Paths, Trails and Circuits in Arc-Colored Digraphs

We deal with different algorithmic questions regarding properly arc-colored s-t paths, trails and circuits in arc-colored digraphs. Given an arc-colored digraph D with c ≥ 2 colors, we show that the problem of maximizing the number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of one properly arc-colored s-t path ...