  This work examines the problem of clique enumeration on a graph by exploiting
its clique covers. The principle of inclusion/exclusion is applied to determine
the number of cliques of size $r$ in the graph union of a set $\mathcal{C} =
\{c_1, \ldots, c_m\}$ of $m$ cliques. This leads to a deeper examination of the
sets involved and to an orbit partition, $\Gamma$, of the power set
$\mathcal{P}(\mathcal{N}_{m})$ of $\mathcal{N}_{m} = \{1, \ldots, m\}$. Applied
to the cliques, this partition gives insight into clique enumeration and yields
new results on cliques within a clique cover, including expressions for the
number of cliques of size $r$ as well as generating functions for the cliques
on these graphs. The quotient graph modulo this partition provides a succinct
representation to determine cliques and maximal cliques in the graph union. The
partition also provides a natural and powerful framework for related problems,
such as the enumeration of induced connected components, by drawing upon a
connection to extremal set theory through intersecting sets.
