  We present exact solutions for the size of the giant connected component
(GCC) of graphs composed of higher-order homogeneous cycles, including weak
cycles and cliques, following bond percolation. We use our theoretical result
to find the location of the percolation threshold of the model, providing
analytical solutions where possible. We expect the results derived here to be
useful to a wide variety of applications including graph theory, epidemiology,
percolation and lattice gas models as well as fragmentation theory. We also
examine the Erd\H{o}s-Gallai theorem as a necessary condition on the
graphicality of configuration model networks comprising higher-order clique
sub-graphs.
