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The Bron-Kerbosch algorithm takes a graph and finds its maximal cliques in an efficient manner (as far as I'm aware, it is $O(3^{n/3})$, where $n$ is the number of vertices).

Let $t$ be a positive integer. I am interested in finding $t$-almost-cliques, that is, induced subgraphs which would be cliques if it wasn't for $t$ or fewer missing edges. Ideally I would like $t$ to depend the size of the almost-clique, but we can fix it for simplicity.

Is there a variant of the Bron-Kerbosch for $t$-almost cliques?

Even a reasonable algorithm would be appreciated.

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  • $\begingroup$ Augment your graph with $t$ edges and apply the Bron-Kerbosch algorithm as a blackbox. $\endgroup$
    – codeR
    Commented Jun 17 at 10:04
  • $\begingroup$ @codeR which $t$ of the $n \choose 2$ edges should be added? $\endgroup$ Commented Jun 17 at 10:29
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    $\begingroup$ One trivial DP Algo has $\mathcal{O}(2^n)$ time complexity, which also gives all the $t$ values for all possible subgraphs $\endgroup$
    – EnEm
    Commented Jun 17 at 11:14
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    $\begingroup$ @codeR Thank you, that's helpful! Additional idea: in the case that $t$ depends on the size of the almost-clique, one can do this for $t=0,1,...,N$ and then filter out the cliques that are missing too many edges in the original graph. It seems to me that this should still be $O(n^{2N}3^{n/3})$, right? $\endgroup$ Commented Jun 17 at 11:15
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    $\begingroup$ @EnEm This was my first thought as well though it seems the augmentation idea might be a bit faster $\endgroup$ Commented Jun 17 at 11:16

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