On the Two Paths Theorem and the Two Disjoint Paths Problem

TL;DR

This paper provides a new, simplified proof of the two paths theorem characterizing 2-linked graphs as webs, and introduces an efficient recursive algorithm for finding two vertex-disjoint paths or embedding graphs into webs.

Contribution

It offers a novel proof avoiding major theorems and develops a practical O(nm) algorithm for the two disjoint paths problem.

Findings

New proof characterizes 2-linked graphs as webs without Kuratowski's or Menger's theorems.

Develops a recursive algorithm with O(nm) complexity for disjoint paths.

Algorithm either finds disjoint paths or embeds the graph into a web.

Abstract

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple proof of the ``two paths theorem'', a characterisation of edge-maximal graphs which are not 2-linked as webs: particular near triangulations filled with cliques. Our proof works by generalising the theorem, replacing the four vertices above by an arbitrary tuple; it does not require major theorems such as Kuratowski's or Menger's theorems. Instead it follows an inductive characterisation of generalised webs via parallel composition, a graph operation consisting in taking a disjoint union before identifying some pairs of vertices. We use the insights provided by this proof to design a simple O(nm) recursive algorithm for the ``two vertex-disjoint paths'' problem. This algorithm is constructive in that it returns either two disjoint paths, or an embedding of the input graph into a web.