Better Quasi Ordering of graphs under the minor relation

Determine whether the quasi-order (Gall, ≤) on finite graphs is a Better Quasi Ordering (BQO), and clarify whether the stronger ω2-WQO property holds; establishing either would have consequences such as the finiteness of second-order obstruction collections for Erdős–Pósa dualities.

Background

The paper discusses how the finiteness of their second-order obstruction collections would follow from stronger quasi-order properties of the minor order on graphs. While Robertson–Seymour proved that (Gall, ≤) is a well-quasi-order (WQO), the stronger ω2-WQO and BQO conjectures remain unsettled.

The authors emphasize that proving these conjectures appears difficult and a constructive proof seems out of reach, but such results would immediately imply finiteness of their obstruction sets for all minor-closed classes H.