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Consider a graph on vertex set $V_1\cup V_2\cup V_3\cup V_4\cup \{a,b,c,d\}$ where $|V_1|=|V_2|=|V_3|=|V_4|=n$. The edge set $E$ is covered by $C=\{V_1\cup\{a,c\},V_2\cup\{a,d\},V_3\cup\{b,c\},V_4\cup\{b,d\},\{a,b\},\{c,d\}\}$.
When $n$ is large enough, any minimalist cover must contain the four maximum cliques $V_1\cup\{a,c\}$ and so on, so it is not hard ...
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I see this question only 2.5 years after, but I think I have a relevant answer. Indeed, it is at the core of the work we have done on Fast generation of random connected graphs with prescribed degrees.
In this paper, we start with a connected graph, and perform large numbers of edge swaps in order to make it random. We however want to obtain a random ...
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This maybe: https://papers-gamma.link/paper/32
With an open source implementation here: https://github.com/maxdan94/kClist
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