A327198 - OEIS

%I #8 Dec 26 2020 23:53:57

%S 0,0,0,1,9,212,9600,789792,114812264,29547629568,13644009626400,

%T 11489505388892800,17918588321874717312,52482523149603539181312,

%U 292311315623259148521270784,3129388799344153886272170009600,64965507855114369076680860799267840

%N Number of labeled simple graphs covering n vertices with vertex-connectivity 2.

%C 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.

%H Andrew Howroyd, <a href="/A327198/b327198.txt">Table of n, a(n) for n = 0..25</a>

%H Gus Wiseman, <a href="/A327198/a327198.png">The a(4) = 9 simple covering graphs with vertex-connectivity 2.</a>

%F a(n) = A013922(n) - A005644(n) for n >= 3. - _Andrew Howroyd_, Dec 26 2020

%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],#]==2&]],{n,0,5}]

%Y The unlabeled version is A052443.

%Y Cf. A005644, A013922, A052442, A259862, A326786, A327082, A327101, A327112, A327113, A327126, A327227.

%K nonn

%O 0,5

%A _Gus Wiseman_, Sep 01 2019

%E Terms a(6) and beyond from _Andrew Howroyd_, Dec 26 2020