The nonlinear selector Picc coincides with the linear Hodge projector PiH on all cochains precisely when the graph is a cactus.
Lim, SIAM Rev.62, 685 (2020)
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The Cactus Criterion: When Nonlinear Hodge Theory Reduces to Linear on Graphs
The nonlinear selector Picc coincides with the linear Hodge projector PiH on all cochains precisely when the graph is a cactus.