Length 3 Edge-Disjoint Paths and Partial Orientation

TL;DR

This paper proves that finding k edge-disjoint paths of length exactly 3 between two vertices is NP-hard, correcting a previous claim of polynomial solvability, by reducing from the NP-hard Partial Orientation problem.

Contribution

It establishes the NP-hardness of the length-3 edge-disjoint paths problem, clarifying the computational complexity and correcting earlier misconceptions.

Findings

Proves the problem is NP-hard even without multiple edges

Reduces from the NP-hard Partial Orientation problem

Clarifies the computational complexity of length-3 edge-disjoint paths

Abstract

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and we show that this problem is NP-hard even if we disallow multiple edges. We use a reduction from Partial Orientation, a problem recently shown by P\'alv\"olgyi to be NP-hard.