A327198 Number of labeled simple graphs covering n vertices with vertex-connectivity 2.

0, 0, 0, 1, 9, 212, 9600, 789792, 114812264, 29547629568, 13644009626400, 11489505388892800, 17918588321874717312, 52482523149603539181312, 292311315623259148521270784, 3129388799344153886272170009600, 64965507855114369076680860799267840

Offset: 0

Author

Gus Wiseman, Sep 01 2019

Keywords

• nonn

Comments

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.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..25
• Gus Wiseman, The a(4) = 9 simple covering graphs with vertex-connectivity 2.

Crossrefs

The unlabeled version is A052443.

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

Programs

Mathematica

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]]]]]]]]];
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]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==2&]],{n,0,5}]

Formula

a(n) = A013922(n) - A005644(n) for n >= 3. - Andrew Howroyd, Dec 26 2020

Extensions

Terms a(6) and beyond from Andrew Howroyd, Dec 26 2020