I have to find number of ways (combination) to create graph that for each 3 vertices there are minimum 2 vertices connected. There is n vertices. For example when n=3, there are 7 possible combinations. How can I make this with maybe dynamic programming?

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    $\begingroup$ Try finding what graphs satisfying this condition look like. $\endgroup$ – Yuval Filmus Feb 10 '18 at 21:26
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    $\begingroup$ Your graphs are the complements of triangle-free graphs. $\endgroup$ – Yuval Filmus Feb 11 '18 at 6:29

Your condition on the graph is the same as requiring its complement to be triangle-free. It is known that there are a lot of triangle-free graph. Indeed, since every bipartite graph is triangle-free, if we fix a partition of the vertex set into two sets of size $n/2$ (assuming for simplicity that $n$ is even), then we already get $2^{n^2/4}$ triangle-free graphs. It turns out that there aren't that much more, see this question on mathoverflow.

Perhaps you'd be interested in a paper on generating maximal triangle-free graphs up to isomorphism (in your case, you will be generating minimal graphs satisfying your condition), whose number is more manageable.

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  • $\begingroup$ Okay, so I need to generate triangle-free graphs and for all of these count combination of complements these graphs? How I can make that for expamle when n=3? (Answer is 7) $\endgroup$ – Sango Feb 19 '18 at 21:49
  • $\begingroup$ Use a software package such as nauty. $\endgroup$ – Yuval Filmus Feb 19 '18 at 21:57
  • $\begingroup$ I cannot use that because of my software, is there any way to generate all possible triangle-free graphs with n vertices? (the best in C++) $\endgroup$ – Sango Feb 19 '18 at 22:11
  • $\begingroup$ I just pointed out a way: use nauty. You can use it both as a standalone program and as a library. $\endgroup$ – Yuval Filmus Feb 19 '18 at 22:15
  • $\begingroup$ Yes, of course but I need to write an algorithm in C++ without any additionally libraries or something like that $\endgroup$ – Sango Feb 19 '18 at 22:19

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