There are two problems for which I would like to find an example that shows the difference between them.
Issue 1 – Minimal Click Edge Coverage: Given a graph $ G $, find the smallest set of cliques covering all the edges in $ G $.
Problem 2 – Maximum cliques: Given a graph $ G $, find the set of maximum cliques in $ G $.
While the solution to problems 1 and 2 guarantees to cover all the edges, I do not understand the difference that exists between them. I understand that the cliques in the solution of problem 1 are not necessarily maximal. Similarly, the set of maximum cliques does not directly imply that their number is minimal, whereas the solution is feasible to the problem 1.
Is there an example that can show me that the solution to these two problems may be different?
I've tried to prove that the solution is not necessarily the same when a triangle ($ C_3 $) overlaps three cliques. But is there an example beyond a triangle?