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

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

Andrew Howroyd, Table of n, a(n) for n = 0..50

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.

Cf. A006129, A054592, A322389, A322390, A326786, A327070, A327098, A327100, A327125, A327126.

Sequence in context: A346315 A058804 A327114 * A355473 A180710 A005328

Adjacent sequences: A327333 A327334 A327335 * A327337 A327338 A327339

Keyword

nonn

Author

Gus Wiseman, Sep 02 2019

Extensions

Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019

Status

approved