  We present several new sufficient conditions for uniform boundedness of the
reduced correlations and free energy of an abstract polymer system in a complex
multidisc around zero fugacity. They resolve a discrepancy between two
incomparable and previously known extensions of Dobrushin's classic condition.
All conditions arise from an extension of the tree-operator approach introduced
by Fern\`{a}ndez & Procacci combined with a novel family of partition schemes
of the spanning subgraph complex of a cluster. The key technique is the
increased transfer of structural information from the partition scheme to a
tree-operator on an enhanced space.
