Finding topological subgraphs is fixed-parameter tractable

TL;DR

This paper proves that testing for a fixed topological subgraph within a larger graph can be done efficiently in polynomial time, resolving longstanding open problems in graph theory and parameterized complexity.

Contribution

It establishes that topological subgraph testing is fixed-parameter tractable with a polynomial-time algorithm for fixed graphs, answering open questions from 1992.

Findings

Topological subgraph testing is fixed-parameter tractable.

Provides a polynomial-time algorithm for fixed graphs.

Answers open questions by Downey and Fellows from 1992.

Abstract

We show that for every fixed undirected graph $H$, there is a $O(|V(G)|^3)$ time algorithm that tests, given a graph $G$, if $G$ contains $H$ as a topological subgraph (that is, a subdivision of $H$ is subgraph of $G$). This shows that topological subgraph testing is fixed-parameter tractable, resolving a longstanding open question of Downey and Fellows from 1992. As a corollary, for every $H$ we obtain an $O(|V(G)|^3)$ time algorithm that tests if there is an immersion of $H$ into a given graph $G$. This answers another open question raised by Downey and Fellows in 1992.