The complexity of the vertex-minor problem
Axel Dahlberg, Jonas Helsen, Stephanie Wehner · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 2022
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. Here we prove that this decision problem is NP-complete, even when restricting H and G to be circle graphs, a class of graphs that has a natural relation to vertex-minors.