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%I #4 Sep 09 2019 12:05:21
%S 0,0,0,72,4752
%N Number of set-systems with n vertices and vertex-connectivity 2.
%C A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
%t csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],vertConnSys[Range[n],#]==2&]],{n,0,3}]
%Y BII-numbers for vertex-connectivity 2 are A327374.
%Y BII-numbers for cut-connectivity 2 are A327082.
%Y BII-numbers for spanning edge-connectivity 2 are A327108.
%Y BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y Labeled graphs with vertex-connectivity 2 are A327198.
%Y The vertex-connectivity of the set-system with BII-number n is A327051(n).
%Y The enumeration of labeled graphs by vertex-connectivity is A327334.
%Y Cf. A013922, A259862, A322389, A322390, A323818, A327336.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Sep 05 2019
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