Biquasiprimitive oriented graphs of valency four

In this short note we describe a recently initiated research programme aiming to use a normal quotient reduction to analyse finite connected, oriented graphs of valency 4, admitting a vertex- and edge-transitive group of automorphisms which
preserves the edge orientation. In the first article on this topic [1], a subfamily of these graphs was identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one ‘basic’ member. These basic members can be further divided into three types: quasiprimitive, biquasiprimitive and cycle type. The first and third of these types were analysed in some detail in the papers [1, 2, 3]. Recently, we have begun an analysis of the basic graphs of biquasiprimitive type. We describe our approach and mention some early results. This work is on-going. It began at the Tutte Memorial MATRIX Workshop.