Complexity classification for three-vertex coloured patterns

Classify the computational complexity of recognizing every graph class expressed by a set of 2-edge-coloured graphs on three vertices.

Background

The paper classifies the complexity of many F-free 2-edge-colouring problems for sets of patterns on at most three vertices, including polynomial-time and NP-complete cases. However, the authors state that the complexity classification is not complete and formulate a problem asking for a classification of every remaining graph class expressed by three-vertex 2-edge-coloured patterns.