A327126 -id:A327126 - cp's OEIS frontend

A327197 Number of set-systems covering n vertices with cut-connectivity 1.

0, 1, 0, 24, 1984

Offset: 0

Gus Wiseman, Sep 01 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain in a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity.

			The a(3) = 24 set-systems:
  {12}{13}  {1}{12}{13}  {1}{2}{12}{13}  {1}{2}{3}{12}{13}
  {12}{23}  {1}{12}{23}  {1}{2}{12}{23}  {1}{2}{3}{12}{23}
  {13}{23}  {1}{13}{23}  {1}{2}{13}{23}  {1}{2}{3}{13}{23}
            {2}{12}{13}  {1}{3}{12}{13}
            {2}{12}{23}  {1}{3}{12}{23}
            {2}{13}{23}  {1}{3}{13}{23}
            {3}{12}{13}  {2}{3}{12}{13}
            {3}{12}{23}  {2}{3}{12}{23}
            {3}{13}{23}  {2}{3}{13}{23}

The BII-numbers of these set-systems are A327098.
The same for cut-connectivity 2 is A327113.
The non-covering version is A327128.
Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327114, A327126, A327229.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
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]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==1&]],{n,0,3}]

Inverse binomial transform of A327128.

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

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.

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

The unlabeled version is A052443.

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

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}]

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

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

A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.

Original entry on oeis.org

0, 1, 2, 27, 2084

Offset: 0

Gus Wiseman, Sep 02 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. We define the cut-connectivity (A326786, A327237, A327126) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

The covering version is A327197.
The BII-numbers of these set-systems are A327098.
Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327113, A327114, A327126, A327229.

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]]]]]]]]];
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]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],cutConnSys[Union@@#,#]==1&]],{n,0,3}]

Binomial transform of A327197.

A327237 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 3, 3, 1, 4, 40, 15, 4, 1, 56, 660, 267, 35, 5, 1, 1031, 18756, 11022, 1862, 90, 6, 1

Offset: 0

Gus Wiseman, Sep 03 2019

We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.

			Triangle begins:
   1
   1   0
   1   0   1
   1   3   3   1
   4  40  15   4   1
  56 660 267  35   5   1

Row sums are A006125.
Column k = 0 is A327199.
The covering case is A327126.
Row sums without the first column are A287689.
Cf. A006125, A001187, A013922, A259862, A322389, A326786, A327070, A327114, A327125, A327127, A327198.

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]]]]]]]]];
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]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],cutConnSys[Union@@#,#]==k&]],{n,0,4},{k,0,n}]

Column-wise binomial transform of A327126.

a(21)-a(27) from Jinyuan Wang, Jun 27 2020

A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 2, 1, 0, 11, 6, 3, 1, 0, 34, 21, 10, 3, 1, 0, 156, 112, 56, 17, 4, 1, 0, 1044, 853, 468, 136, 25, 4, 1, 0, 12346, 11117, 7123, 2388, 384, 39, 5, 1, 0, 274668, 261080, 194066, 80890, 14480, 1051, 59, 5, 1, 0, 12005168, 11716571, 9743542, 5114079, 1211735, 102630, 3211, 87, 6, 1, 0

Offset: 0

Gus Wiseman, Sep 26 2019

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. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
   1
   1  0
   2  1  0
   4  2  1  0
  11  6  3  1  0
  34 21 10  3  1  0

Gus Wiseman, The graphs counted in row n = 4 (isolated vertices not shown).

Row-wise partial sums of A259862.
The labeled version is A327363.
The covering case is A327365, from which this sequence differs only in the k = 0 column.
Column k = 0 is A000088 (graphs).
Column k = 1 is A001349 (connected graphs), if we assume A001349(0) = A001349(1) = 0.
Column k = 2 is A002218 (2-connected graphs), if we assume A002218(2) = 0.
The triangle for vertex-connectivity exactly k is A259862.
Cf. A326786, A327051, A327114, A327125, A327126, A327127, A327334.

T(n,k) = Sum_{j=k..n} A259862(n,j).

Terms a(21) and beyond from Andrew Howroyd, Dec 26 2020

A327234 Smallest BII-number of a set-system with cut-connectivity n.

Original entry on oeis.org

0, 1, 4, 52, 2868

Offset: 0

Gus Wiseman, Sep 03 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
We define the cut-connectivity (A326786) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).
Conjecture: a(n > 1) = A327373(n) = the BII-number of K_n.

			The sequence of terms together with their corresponding set-systems:
     0: {}
     1: {{1}}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

The same for spanning edge-connectivity is A327147.
The cut-connectivity of the set-system with BII-number n is A326786(n).
Cf. A000120, A002450, A029931, A048793, A070939, A259862, A326031, A327082, A327098, A327125, A327126, A327127, A327373.

A327363 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 8, 4, 1, 0, 64, 38, 10, 1, 0, 1024, 728, 238, 26, 1, 0

Offset: 0

Gus Wiseman, Sep 26 2019

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.

			Triangle begins:
     1
     1    0
     2    1    0
     8    4    1    0
    64   38   10    1    0
  1024  728  238   26    1    0

Column k = 0 is A006125.
Column k = 1 is A001187.
Column k = 2 is A013922.
The unlabeled version is A327805.
Row-wise partial sums of A327334 (vertex-connectivity exactly k).
Cf. A259862, A327114, A327125, A327126, A327127, A327806.

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],#]>=k&]],{n,0,4},{k,0,n}]

cp's OEIS Frontend

A327197 Number of set-systems covering n vertices with cut-connectivity 1.

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

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A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.

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A327237 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.

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A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

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A327234 Smallest BII-number of a set-system with cut-connectivity n.

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A327363 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity >= k.

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Mathematica