A322395 -id:A322395 - cp's OEIS frontend

A095983 Number of 2-edge-connected labeled graphs on n nodes.

0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016

Offset: 0

Yifei Chen (yifei(AT)mit.edu), Jul 17 2004

nonn

From Falk Hüffner, Jun 28 2018: (Start)
Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are k-edge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
Labeled version of A007146. (End)
The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. This sequence counts graphs with spanning edge-connectivity >= 2, which, for n > 1, are connected graphs with no bridges. - Gus Wiseman, Sep 20 2019

Jinyuan Wang, Table of n, a(n) for n = 0..82
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276-305.
Gus Wiseman, The a(4) = 10 labeled graphs with spanning edge-connectivity >= 2.
Gus Wiseman, Non-isomorphic representatives of the a(5) = 253 graphs with spanning edge-connectivity >= 2.

Cf. A053549, A198046.
The unlabeled version is A007146.
Row sums of A327069 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with spanning edge-connectivity 2 are A327146.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Graphs without endpoints are A059167.
Graphs with spanning edge-connectivity 1 are A327071.
Cf. A001187, A002218, A052446, A059166, A100743, A322395, A327079, A327144.

seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k-1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
seq[16] (* Jean-François Alcover, Aug 13 2019, after Andrew Howroyd *)
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]>=2&]],{n,0,5}] (* Gus Wiseman, Sep 20 2019 *)

\\ here p is initially A053549, q is A198046 as e.g.f.s.
seq(n)={my(v=vector(n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
my(q=x*exp(p)); p-=q;
for(k=3, n, my(c=polcoeff(p,k)); v[k]=c*(k-1)!; p-=c*q^k);
concat([0],v)} \\ Andrew Howroyd, Jun 18 2018

seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x-1), -(n+1))} \\ Andrew Howroyd, Dec 28 2020

a(n) = A001187(n) - A327071(n). - Gus Wiseman, Sep 20 2019

Name corrected and more terms from Pavel Irzhavski, Nov 01 2014
Offset corrected by Falk Hüffner, Jun 17 2018
a(12)-a(16) from Andrew Howroyd, Jun 18 2018

A327071 Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.

Original entry on oeis.org

0, 0, 1, 3, 28, 475, 14736, 818643, 82367552, 15278576679, 5316021393280, 3519977478407687, 4487518206535452672, 11116767463976825779115, 53887635281876408097483776, 513758302006787897939587736715, 9668884580476067306398361085853696

Offset: 0

Gus Wiseman, Aug 24 2019

nonn

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).

The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty graph.

Jean-François Alcover and Vaclav Kotesovec, Table of n, a(n) for n = 0..82 [using A001187 and b-file from A095983]
Eric Weisstein's World of Mathematics, Bridged Graph

Column k = 1 of A327069.
The unlabeled version is A052446.
Connected graphs without bridges are A007146.
The enumeration of labeled connected graphs by number of bridges is A327072.
Connected graphs with exactly one bridge are A327073.
Graphs with non-spanning edge-connectivity 1 are A327079.
BII-numbers of set-systems with spanning edge-connectivity 1 are A327111.
Covering set-systems with spanning edge-connectivity 1 are A327145.
Graphs with spanning edge-connectivity 2 are A327146.
Cf. A001187, A001349, A006125, A059166, A322395, A327071, A327077, A327099, A327144.

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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]==1&]],{n,0,4}]

a(1) = 0; a(n > 1) = A001187(n) - A095983(n).

A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Aug 25 2019

nonn

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.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  32: {{2,3}}

Graphs with spanning edge-connectivity >= 2 are counted by A095983.
BII-numbers for vertex-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
Graphs with spanning edge-connectivity 1 are counted by A327145.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

A059166 Number of n-node connected labeled graphs without endpoints.

Original entry on oeis.org

1, 1, 0, 1, 10, 253, 12058, 1052443, 169488200, 51045018089, 29184193354806, 32122530765469967, 68867427921051098084, 290155706369032525823085, 2417761578629525173499004146, 40013923790443379076988789688611, 1318910080173114018084245406769861936

Offset: 0

Vladeta Jovovic, Jan 12 2001

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 404.

Vaclav Kotesovec, Table of n, a(n) for n = 0..80

Cf. A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).

Cf. A001187, A007146, A095983, A100743, A261919, A322395.

c:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*c(k), k=1..n-1)/n)
    end:
a:= n-> max(0, add((-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, i=0..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 27 2017

Flatten[{1,1,0,Table[n!*Sum[(-1)^(n-j)*SeriesCoefficient[1+Log[Sum[2^(k*(k-1)/2)*x^k/k!,{k,0,j}]],{x,0,j}]*j^(n-j)/(n-j)!,{j,0,n}],{n,3,15}]}] (* Vaclav Kotesovec, May 14 2015 *)
c[0] = 1; c[n_] := c[n] = 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k)*(n - k - 1)/2)*c[k], {k, 1, n-1}]/n; a[0] = a[1] = 1; a[2] = 0; a[n_] := Sum[(-1)^i*Binomial[n, i]*c[n-i]*(n-i)^i, {i, 0, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 2017, using Alois P. Heinz's code for c(n) *)

seq(n)={Vec(serlaplace(1 + x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))))} \\ Andrew Howroyd, Sep 09 2018

a(n) = Sum_{i=0..n} (-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, for n>2, a(0)=1, a(1)=1, a(2)=0, where c(n) is number of n-node connected labeled graphs (cf. A001187).
E.g.f.: 1 + x^2/2 + log(Sum_{n >= 0} 2^binomial(n, 2)*(x*exp(-x))^n/n!).
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, May 14 2015
Logarithmic transform of A100743, if we assume a(1) = 0. - Gus Wiseman, Aug 15 2019

More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001

A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 4, 18, 27, 14, 1, 56, 250, 402, 240, 65, 10, 1, 1031, 5475, 11277, 9620, 4282, 921, 146, 15, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any isolated vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   1
   1   1
   1   3   3   1
   4  18  27  14   1
  56 250 402 240  65  10   1

Row sums are A006125.
Column k = 0 is A327199.
Column k = 1 is A327231.
The corresponding triangle for vertex-connectivity is A327125.
The corresponding triangle for spanning edge-connectivity is A327069.
The covering version is A327149.
The unlabeled version is A327236, with covering version A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.

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]]]]]]]]];
edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.

a(20)-a(28) from Robert Price, May 25 2021

A100743 Number of labeled n-vertex graphs without vertices of degree <=1.

Original entry on oeis.org

1, 0, 0, 1, 10, 253, 12068, 1052793, 169505868, 51046350021, 29184353055900, 32122563615242615, 68867440268165982320, 290155715157676330952559, 2417761590648159731258579164, 40013923822242935823157820555477, 1318910080336893719646370269435043184

Offset: 0

Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 03 2005

nonn

			From _Gus Wiseman_, Aug 18 2019: (Start)
The a(4) = 10 edge-sets:
  {12,13,24,34}
  {12,14,23,34}
  {13,14,23,24}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
(End)

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

Graphs without isolated nodes are A006129.
The connected case is A059166.
Graphs without endpoints are A059167.
Labeled graphs with endpoints are A245797.
The unlabeled version is A261919.
Cf. A095983, A136284, A322395, A327079, A327107, A327227, A327229, A327230.
Cf. A095983, A322395.

m = 13;
egf = Exp[-x + x^2/2]*Sum[2^(n (n-1)/2)*(x/Exp[x])^n/n!, {n, 0, m+1}];
s = egf + O[x]^(m+1);
a[n_] := n!*SeriesCoefficient[s, n];
Table[a[n], {n, 0, m}] (* Jean-François Alcover, Feb 23 2019 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}] (* Gus Wiseman, Aug 18 2019 *)

seq(n)={Vec(serlaplace(exp(-x + x^2/2 + O(x*x^n))*sum(k=0, n, 2^(k*(k-1)/2)*(x/exp(x + O(x^n)))^k/k!)))} \\ Andrew Howroyd, Sep 04 2019

E.g.f.: exp(-x+x^2/2)*(Sum_{n>=0} 2^(n*(n-1)/2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Jan 26 2006
Exponential transform of A059166. - Gus Wiseman, Aug 18 2019
Inverse binomial transform of A059167. - Gus Wiseman, Sep 02 2019

Terms a(14) and beyond from Andrew Howroyd, Sep 04 2019

A327079 Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080

Offset: 0

Gus Wiseman, Aug 25 2019

nonn

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Graphs with no bridges are counted by A095983 (2-edge-connected graphs).

Also labeled simple connected graphs covering n vertices with non-spanning edge-connectivity 1, where the non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

Column k = 1 of A327149.
The non-covering version is A327231.
Connected bridged graphs (spanning edge-connectivity 1) are A327071.
BII-numbers of graphs with non-spanning edge-connectivity 1 are A327099.
Covering set-systems with non-spanning edge-connectivity 1 are A327129.
Cf. A001187, A006129, A052446, A059166, A322395, A327072, A327073, A327148.

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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,4}]

a(n) = A001187(n) - A322395(n) for n > 2. - Andrew Howroyd, Aug 27 2019

Inverse binomial transform of A327231.

Terms a(6) and beyond from Andrew Howroyd, Aug 27 2019

A327097 BII-numbers of set-systems with non-spanning edge-connectivity 2.

Original entry on oeis.org

5, 6, 17, 20, 24, 34, 36, 40, 48, 53, 54, 55, 60, 61, 62, 63, 65, 66, 68, 71, 72, 80, 86, 87, 89, 92, 93, 94, 95, 96, 101, 103, 106, 108, 109, 110, 111, 113, 114, 115, 121, 122, 123, 257, 260, 272, 308, 309, 310, 311, 316, 317, 318, 319, 320, 326, 327, 342

Offset: 1

Gus Wiseman, Aug 20 2019

nonn

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.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any isolated vertices) to result in a disconnected or empty set-system.

			The sequence of all set-systems with non-spanning edge-connectivity 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  24: {{3},{1,3}}
  34: {{2},{2,3}}
  36: {{1,2},{2,3}}
  40: {{3},{2,3}}
  48: {{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  63: {{1},{2},{1,2},{3},{1,3},{2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}

Positions of 2's in A326787.
BII-numbers for vertex-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for non-spanning edge-connectivity > 1 are A327102.
BII-numbers for spanning edge-connectivity 2 are A327108.
Cf. A007146, A048793, A052446, A059166, A070939, A095983, A263296, A322335, A322338, A322395, A326031, A327041, A327069, A327111.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]==2&]

A327102 BII-numbers of set-systems with non-spanning edge-connectivity >= 2.

Original entry on oeis.org

5, 6, 17, 20, 21, 24, 34, 36, 38, 40, 48, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

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.

A set-system has non-spanning 2-edge-connectivity >= 2 if it is connected and any single edge can be removed (along with any non-covered vertices) without making the set-system disconnected or empty. Alternatively, these are connected set-systems whose bridges (edges whose removal disconnects the set-system or leaves isolated vertices) are all endpoints (edges intersecting only one other edge).

			The sequence of all set-systems with non-spanning edge-connectivity >= 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  24: {{3},{1,3}}
  34: {{2},{2,3}}
  36: {{1,2},{2,3}}
  38: {{2},{1,2},{2,3}}
  40: {{3},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  56: {{3},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  63: {{1},{2},{1,2},{3},{1,3},{2,3}}

Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with non-spanning edge-connectivity >= 2 are counted by A322395.
Also positions of terms >=2 in A326787.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108, A327148.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]>=2&]

A327129 Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 1, 2, 35, 2804

Offset: 0

Gus Wiseman, Aug 27 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			The a(3) = 35 set-systems:
  {123}  {1}{12}{23}   {1}{2}{12}{13}   {1}{2}{3}{12}{13}
         {1}{13}{23}   {1}{2}{12}{23}   {1}{2}{3}{12}{23}
         {1}{2}{123}   {1}{2}{13}{23}   {1}{2}{3}{13}{23}
         {1}{3}{123}   {1}{2}{3}{123}   {1}{2}{3}{12}{123}
         {2}{12}{13}   {1}{3}{12}{13}   {1}{2}{3}{13}{123}
         {2}{13}{23}   {1}{3}{12}{23}   {1}{2}{3}{23}{123}
         {2}{3}{123}   {1}{3}{13}{23}
         {3}{12}{13}   {2}{3}{12}{13}
         {3}{12}{23}   {2}{3}{12}{23}
         {1}{23}{123}  {2}{3}{13}{23}
         {2}{13}{123}  {1}{2}{13}{123}
         {3}{12}{123}  {1}{2}{23}{123}
                       {1}{3}{12}{123}
                       {1}{3}{23}{123}
                       {2}{3}{12}{123}
                       {2}{3}{13}{123}

The restriction to simple graphs is A327079, with non-covering version A327231.
The version for spanning edge-connectivity is A327145, with BII-numbers A327111.
The BII-numbers of these set-systems are A327099.
The non-covering version is A327196.
Cf. A003465, A006129, A263296, A322395, A323818, A327071, A327149.

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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,3}]

Inverse binomial transform of A327196.

cp's OEIS Frontend

A095983 Number of 2-edge-connected labeled graphs on n nodes.

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A327071 Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.

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A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

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A059166 Number of n-node connected labeled graphs without endpoints.

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A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

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A100743 Number of labeled n-vertex graphs without vertices of degree <=1.

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A327079 Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).

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A327097 BII-numbers of set-systems with non-spanning edge-connectivity 2.