A New Graph Construction of Unbounded Clique-width

We define permutation-partition graphs by replacing one part of a 2K2-free bipartite graph (a bipartite chain graph) by an induced linear forest. We show that this hereditary graph class is of of unbounded clique-width (with a new graph construction of large clique-width). We show that this graph class contains no minimal graph class of unbounded clique-width, and give a conjecture for a contained boundary class for this property.