A Linear Time Algorithm for Finding Three Edge-Disjoint Paths in Eulerian Networks

Abstract

Consider an undirected graph G = (VG, EG) and a set of six terminals T = {s ₁, s ₂, s ₃, t ₁, t ₂, t ₃} ⊆ VG. The goal is to find a collection $\mathcal{P}$ of three edge-disjoint paths P ₁, P ₂, and P ₃, where P _i  connects nodes s _i and t _i (i = 1, 2, 3).

Results obtained by Robertson and Seymour by graph minor techniques imply a polynomial time solvability of this problem. The time bound of their algorithm is O(m ³) (hereinafter we assume n : = |VG|, m : = |EG|, n = O(m)).

In this paper we consider a special, Eulerian case of G and T. Namely, construct the demand graph H = (VG, {s ₁ t ₁, s ₂ t ₂, s ₃ t ₃}). The edges of H correspond to the desired paths in $\mathcal{P}$. In the Eulerian case the degrees of all nodes in the (multi-) graph G + H (= (VG, EG ∪ EH)) are even.

Schrijver showed that, under the assumption of Eulerianess, cut conditions provide a criterion for the existence of $\mathcal{P}$. This, in particular, implies that checking for existence of $\mathcal{P}$ can be done in O(m) time. Our result is a combinatorial O(m)-time algorithm that constructs $\mathcal{P}$ (if the latter exists).