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%I #8 Sep 11 2019 17:38:28
%S 0,0,1,3,28,490,15336,851368,85010976,15615858960,5388679220480,
%T 3548130389657216,4507988483733389568,11145255551131555572992,
%U 53964198507018134569758720,514158235191699333805861463040,9672967865350359173180572164444160
%N Number of labeled simple graphs with vertex-connectivity 1.
%C Same as A327114 except a(2) = 1.
%C The vertex-connectivity of a graph is the minimum number of vertices that must be removed (along with any incident edges) to obtain a non-connected graph or singleton.
%H Andrew Howroyd, <a href="/A327336/b327336.txt">Table of n, a(n) for n = 0..50</a>
%e The a(2) = 1 through a(4) = 28 edge-sets:
%e {12} {12,13} {12,13,14}
%e {12,23} {12,13,24}
%e {13,23} {12,13,34}
%e {12,14,23}
%e {12,14,34}
%e {12,23,24}
%e {12,23,34}
%e {12,24,34}
%e {13,14,23}
%e {13,14,24}
%e {13,23,24}
%e {13,23,34}
%e {13,24,34}
%e {14,23,24}
%e {14,23,34}
%e {14,24,34}
%e {12,13,14,23}
%e {12,13,14,24}
%e {12,13,14,34}
%e {12,13,23,24}
%e {12,13,23,34}
%e {12,14,23,24}
%e {12,14,24,34}
%e {12,23,24,34}
%e {13,14,23,34}
%e {13,14,24,34}
%e {13,23,24,34}
%e {14,23,24,34}
%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],{2}]],vertConnSys[Range[n],#]==1&]],{n,0,4}]
%Y Column k = 1 of A327334.
%Y The unlabeled version is A052442.
%Y Connected non-separable graphs are A013922.
%Y Set-systems with vertex-connectivity 1 are A327128.
%Y Labeled simple graphs with cut-connectivity 1 are A327114.
%Y Cf. A006129, A054592, A322389, A322390, A326786, A327070, A327098, A327100, A327125, A327126.
%K nonn
%O 0,4
%A _Gus Wiseman_, Sep 02 2019
%E Terms a(6) and beyond from _Andrew Howroyd_, Sep 11 2019
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