From edge-disjoint paths to independent paths

TL;DR

This paper establishes the relationship between edge-disjoint s-t paths and independent u-v paths in simple undirected graphs, showing that the maximum number of independent paths is equal to the number of edge-disjoint paths for k<3 and 3 for larger k.

Contribution

The paper proves a precise characterization of the maximum independent u-v paths in relation to edge-disjoint s-t paths, resolving a specific graph theory problem.

Findings

f(k)=k for k<3

f(k)=3 for k≥3

establishes a clear relationship between edge-disjoint and independent paths

Abstract

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<3, and f(k)=3 otherwise.