The Condition of Beineke and Harary on Edge-Disjoint Paths Some of Which are Openly Disjoint

4.1 Edge Disjoint Paths

Problem Statement: Given a directed graphG and a set of terminal pairs {(s1, t1), (s2, t2), · · · , (sk, tk)}, our goal is to connect as many pairs as possible using non edge intersecting paths. Edge disjoint paths problem is NP-Complete and is closely related to the multicommodity flow problem. In fact integer multicommodity flow is a generalization of this problem. We describe a greedy approx...

Reconstructing edge-disjoint paths

Reconstructing edge-disjoint paths faster

For a simple undirected graph with n vertices and m edges, we consider a data structure that given a query of a pair of vertices u, v and an integer k ≥ 1, it returns k edge-disjoint uv-paths. The data structure takes Õ(n) time to build, using O( √ mn log n) space, and each query takes O( √ kn) time, which is optimal and beats the previous query time of O(knα(n)).

Edge disjoint Paths in Graphs on Surfaces

We present a survey of results on the edge disjoint paths problem and re late this problem to the edge disjoint homotopic path and the edge disjoint homotopic cycle problem The latter problem is given a graph G V E embedded on a surface S and closed curves C Ck on S nd necessary and su cient conditions for the existence of pairwise edge disjoint cycles e C e Ck in G so that e Ci is homotopic to...

From edge-disjoint paths to independent paths

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s, t and k edge-disjoint s–t paths, also contains two vertices u, v and f(k) independent u–v paths. Here, a set of paths is independent if none of them contains an interior vertex of another. We prove that f(k) = ( k if k ≤ 2, and 3 otherwise. Since independent paths are edge-disjoint, it is clear that f...