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%I #15 Sep 11 2019 17:36:48
%S 0,0,0,3,28,490,15336,851368,85010976,15615858960,5388679220480,
%T 3548130389657216,4507988483733389568,11145255551131555572992,
%U 53964198507018134569758720,514158235191699333805861463040
%N Number of labeled simple graphs covering n vertices with cut-connectivity 1.
%C The cut-connectivity of a graph is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a disconnected or empty graph.
%H Andrew Howroyd, <a href="/A327114/b327114.txt">Table of n, a(n) for n = 0..50</a>
%F a(n) = A001187(n) - A013922(n), if we assume A001187(1) = 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 cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==1&]],{n,0,3}]
%o (PARI) seq(n)={my(g=log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))); Vec(serlaplace(g-intformal(1+log(x/serreverse(x*deriv(g))))), -(n+1))} \\ _Andrew Howroyd_, Sep 11 2019
%Y Column k = 1 of A327126.
%Y The unlabeled version is A052442, if we assume A052442(2) = 0.
%Y Connected non-separable graphs are A013922.
%Y BII-numbers for cut-connectivity 1 are A327098.
%Y Set-systems with cut-connectivity 1 are counted by A327197.
%Y Labeled simple graphs with vertex-connectivity 1 are A327336.
%Y Cf. A001187, A054592, A259862, A322389, A322390, A326786, A327070, A327100, A327125.
%K nonn
%O 0,4
%A _Gus Wiseman_, Aug 25 2019
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