Fig. 1: Edge-disjoint edge clique cover (EECC) of a small simplicial complex (SC).

The SC, shown on the left, consists of fourteen nodes (0-simplices), identified via letters, connected by one 3-simplex, one 2-simplex, and fourteen 1-simplices. Gray areas indicate r-simplices with r ⩾ 2, including r + 1 nodes each. The EECC of the SC is shown on the right, where colored and dotted areas are used to visualize, respectively, the (1, r)-cliques and (0, r)-cliques in it, with colors carrying no specific meanings. The SC is decomposed in: one (1, 4)-clique, {b, d, m, n}; one (1, 3)-clique, {i, j, k}; three (0, 3)-cliques, {c, d, e}, {f, g, m}, {h, i, m}; and five (0, 2)-cliques, {a, b}, {a, n}, {g, h}, {k, l}, {l, m}. The underlying subgraph induced by the subset {a, b, d, m, n} originally consists of a (1, 4)-clique and a (0, 3)-clique. To preserve the group interaction mediated by the (1, 4)-clique, is preferable to include this in the EECC and then break the (0, 3)-clique into two (non-maximal) (0, 2)-cliques. Besides, the underlying subgraph induced by the subset {f, g, h, i, m} is made of three overlapping (0, 3)-cliques, and the EECC is in this case obtained by including {f, g, m} and {h, i, m} (and then the remaining edge {g, h}), instead of {g, h, m} first.