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A327336 Number of labeled simple graphs with vertex-connectivity 1.

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