On the Two Paths Theorem and the Two Disjoint Paths Problem

2025-05-22

AI Summary

This paper addresses the 2-linkedness decision problem for undirected graphs: given a graph and two vertex pairs $(s_1,t_1)$ and $(s_2,t_2)$, does there exist a pair of vertex-disjoint $s_1$–$t_1$ and $s_2$–$t_2$ paths? The central objective is to provide a concise, constructive proof of the “Two Paths Theorem”, characterizing edge-maximal non-2-linked graphs as generalized web graphs. Methodologically, we introduce an inductive characterization of web graphs and a parallel composition construction, extending the four-terminal case to arbitrary terminal tuples and establishing a novel proof framework independent of Kuratowski’s or Menger’s theorems. We further design a recursive algorithm based on web-embedding detection, achieving $O(nm)$ time complexity for constructive 2-linkedness testing—capable of either outputting two disjoint paths or producing an explicit web embedding certifying non-2-linkedness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Linked Open Data, Knowledge Graphs & KB Completion
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the 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.

Problem

• Characterize edge-maximal non-2-linked graphs as webs
• Provide inductive proof for generalized webs via parallel composition
• Design O(nm) algorithm for vertex-disjoint paths problem

Innovation

• New proof for Two Paths Theorem
• Generalizes theorem with arbitrary tuples
• Recursive O(nm) algorithm for paths