A Unified Erdős–Pósa Theorem for Constrained Cycles
A (Γ 1 ,Γ 2 )-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups Γ 1 ,Γ 2 . A cycle in such a labeled graph is (Γ 1 ,Γ 2 )-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to (Γ 1 ,Γ 2 )-labeled graphs. As an application, we determine all canonical obstructions to the Erdős–Pósa property for (Γ 1 ,Γ 2 )-non-zero cycles in (Γ 1 ,Γ 2 )-labeled graphs.