  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. They provide the only examples discussed by Britto and collaborators so
far. We call such graphs simple graphs. The Dunce's cap graph is the first
non-simple graph. The number of its spanning trees (five) exceeds the number of
its edges (four). We compare the two coactions which indeed do not agree and
discuss this result. We also point out that for kinematic renormalization
schemes the coaction $\Delta^G$ simplifies.
