  A decomposition of a chemical reaction network (CRN) is produced by
partitioning its set of reactions. The partition induces networks, called
subnetworks, that are "smaller" than the given CRN which, at this point, can be
called parent network. A complex is called a common complex if it occurs in at
least two subnetworks in a decomposition. A decomposition is said to be
incidence independent if the image of the incidence map of the parent network
is the direct sum of the images of the subnetworks' incidence maps. It has been
recently discovered that the complex balanced equilibria of the parent network
and its subnetworks are fundamentally connected in an incidence independent
decomposition. In this paper, we utilized the set of common complexes and a
developed criterion to investigate decomposition's incidence independence
properties. A framework was also developed to analyze decomposition classes
with similar structure and incidence independence properties. We identified
decomposition classes that can be characterized by their sets of common
complexes and studied their incidence independence. Some of these decomposition
classes occur in some biological and chemical models. Finally, a sufficient
condition was obtained for the complex balancing of some power law kinetic
(PLK) systems with incidence independent and complex balanced decompositions.
This condition led to a generalization of the Defficiency Zero Theorem for some
PLK systems.
