# The complexity of the vertex-minor problem

## TL;DR

This paper proves that determining whether a graph has a specific vertex-minor is NP-complete, even within the class of circle graphs, highlighting the computational difficulty of problems in this area.

## Contribution

It establishes the NP-completeness of the vertex-minor decision problem for circle graphs, a significant class related to vertex-minors, which was previously unknown.

## Key findings

- Deciding vertex-minors is NP-complete.
- NP-completeness holds even for circle graphs.
- Highlights computational challenges in graph theory and quantum information applications.

## Abstract

A graph H is a vertex-minor of a graph G if it can be reached from G by the successive application of local complementations and vertex deletions. Vertex-minors have been the subject of intense study in graph theory over the last decades and have found applications in other fields such as quantum information theory. Therefore it is natural to consider the computational complexity of deciding whether a given graph G has a vertex-minor isomorphic to another graph H, which was previously unknown. Here we prove that this decision problem is NP-complete, even when restricting H, G to be circle graphs, a class of graphs that has a natural relation to vertex-minors.