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A327336
Number of labeled simple graphs with vertex-connectivity 1.
12
0, 0, 1, 3, 28, 490, 15336, 851368, 85010976, 15615858960, 5388679220480, 3548130389657216, 4507988483733389568, 11145255551131555572992, 53964198507018134569758720, 514158235191699333805861463040, 9672967865350359173180572164444160
OFFSET
0,4
COMMENTS
Same as A327114 except a(2) = 1.
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.
LINKS
EXAMPLE
The a(2) = 1 through a(4) = 28 edge-sets:
{12} {12,13} {12,13,14}
{12,23} {12,13,24}
{13,23} {12,13,34}
{12,14,23}
{12,14,34}
{12,23,24}
{12,23,34}
{12,24,34}
{13,14,23}
{13,14,24}
{13,23,24}
{13,23,34}
{13,24,34}
{14,23,24}
{14,23,34}
{14,24,34}
{12,13,14,23}
{12,13,14,24}
{12,13,14,34}
{12,13,23,24}
{12,13,23,34}
{12,14,23,24}
{12,14,24,34}
{12,23,24,34}
{13,14,23,34}
{13,14,24,34}
{13,23,24,34}
{14,23,24,34}
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], #]==1&]], {n, 0, 4}]
CROSSREFS
Column k = 1 of A327334.
The unlabeled version is A052442.
Connected non-separable graphs are A013922.
Set-systems with vertex-connectivity 1 are A327128.
Labeled simple graphs with cut-connectivity 1 are A327114.
Sequence in context: A346315 A058804 A327114 * A355473 A180710 A005328
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 02 2019
EXTENSIONS
Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019
STATUS
approved