Outer Space as a combinatorial backbone for Cutkosky rules and coactions

Understand

We consider a coaction which exists for any bridge-free graph.

• It is based on the cubical chain complex associated to any such graph by considering two boundary operations: shrinking edges or removing them.
• Only if the number of spanning trees of a graph $G$ equals its number of internal edges we find that the graphical coaction $\Delta^G$ constructed here agrees with the coaction $\Delta_{\mathsf{Inc}}$ proposed by Britto and collaborators.
• The graphs for which this is the case are one-loop graphs or their duals, multi-edge banana graphs.