Vertex‐disjoint paths and edge‐disjoint branchings in directed graphs

Abstract

A theorem of J. Edmonds states that a directed graph has k edge-disjoint branchings rooted at a vertex r if and only if every vertex has k edge-disjoint paths to r . We conjecture an extension of this theorem to vertex-disjoint paths and give a constructive proof of the conjecture in the case k = 2. THE CONJECTURELet G = ( V , E ) be a finite directed graph with vertex set V and edge set E . Multiple edges are allowed, but self loops are excluded. An edge directed from x to y will be denoted by (x,y) (we do not distinguish between multiple edges from x to y), and we refer to x as the tail and y as the head of this edge. If P is a directed path from x to y , we refer to x and y as the tail and head of P , respectively, and say that P is trivial if x and y are the same vertex. If x' and y ' are the tail and head of a subpath of P , we write P :x' + y ' for the restriction of P to this path. Two paths are called edge-disjoint if they have no common edge and vertex-disjoint if they have no common vertices except possibly a common head or tail (the trivial path x is vertex-disjoint from precisely those paths with head or tail x). For a subset R of V , an R-branching in G is a spanning forest B of G in which all vertices of V -R have outdegree precisely 1. When R just consists of a single vertex r, we refer to B as an r-branching.Let I = (1, . . . , k}, where 1 5 k 5 IVI. Let R = { R, 1 i E I } be a family of subsets of V, and let B = {B, I i E I } be a family of edge-disjoint branchings in G such that B, is an R,-branching. Let X = {x, I i E I } be a family of (possibly

Output-Sensitive Reporting of Disjoint Paths

“…Similar conjectures have been formulated considering edge-connectivity instead of vertex-connectivity [27], [34] and for directed graphs [17], [25], [49], [54].…”

Arborescence Problems in Directed Graphs: Theorems and Algorithms

“…3. On the other hand, Whitty [49] proved that the statement is true in the case of k 2. Furthermore, Huck [29] proved that if D is acyclic, then the statement is true.…”

“…D is acyclic and 8v 2 V: jqðvÞj 2: ð3:12Þ Theorem 3.9 in the cases ð3.10Þ and ð3.11Þ can be regarded as generalizations of the results of Whitty [49] and Huck [29], respectively, by using convex sets. On the other hand, Theorem 3.9 in the case ð3.12Þ can be regarded as a partial generalization of the result of Huck [29] to the multiple roots case.…”

Directeds-t numberings, Rubber bands, and testing digraphk-vertex connectivity

“…Previously, a much more complicated construction has been reported by Whitty [24]; very recently, we have learned of another construction due to Plehn [21]. Previously, a much more complicated construction has been reported by Whitty [24]; very recently, we have learned of another construction due to Plehn [21].…”