  The vector space of all polygons with configurations of diagonals is endowed
with an operad structure. This is the consequence of a functorial construction
$\mathsf{C}$ introduced here, which takes unitary magmas $\mathcal{M}$ as input
and produces operads. The obtained operads involve regular polygons with
configurations of arcs labeled on $\mathcal{M}$, called $\mathcal{M}$-decorated
cliques and generalizing usual polygons with configurations of diagonals. We
provide here a complete study of the operads $\mathsf{C}\mathcal{M}$. By
considering combinatorial subfamilies of $\mathcal{M}$-decorated cliques
defined, for instance, by limiting the maximal number of crossing diagonals or
the maximal degree of the vertices, we obtain suboperads and quotients of
$\mathsf{C}\mathcal{M}$. This leads to a new hierarchy of operads containing,
among others, operads on noncrossing configurations, Motzkin configurations,
forests, dissections of polygons, and involutions. We show that the suboperad
of noncrossing configurations is Koszul and exhibit its presentation by
generators and relations. Besides, the construction $\mathsf{C}$ leads to
alternative definitions of several operads, like the operad of bicolored
noncrossing configurations and the operads of simple and double multi-tildes.
