A327097 -id:A327097 - cp's OEIS frontend

A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

0, 0, 0, 4, 42, 718, 26262, 1878422, 256204460, 67525498676, 34969833809892, 35954978661632864, 73737437034063350534, 302166248212488958298674, 2475711390267267917290354410, 40563960064630744031043287569378, 1329219366981359393514586291328267704

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

Gus Wiseman, The a(4) = 42 graphs with non-spanning edge-connectivity >= 2.

Row sums of A327148 if the first two columns are removed.
BII-numbers of set-systems with non-spanning edge-connectivity >= 2 are A327102.
Graphs with non-spanning edge-connectivity 1 are A327231.
Cf. A001187, A006129, A095983, A182100, A322395, A326787, A327076, A327079, A327097, A327099, A327236.

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]]]]]]]]];
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}]],eConn[#]>=2&]],{n,0,5}]

Binomial transform of A322395, if we assume A322395(0) = A322395(1) = A322395(2) = 0.

A327374 BII-numbers of set-systems with vertex-connectivity 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Gus Wiseman, Sep 04 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 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.

			The sequence of all set-systems with vertex-connectivity 2 together with their BII-numbers begins:
  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}}
  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}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  67: {{1},{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}
  73: {{1},{3},{1,2,3}}
  74: {{2},{3},{1,2,3}}
  75: {{1},{2},{3},{1,2,3}}

Positions of 2's in A327051.
Cut-connectivity 2 is A327082.
Spanning edge-connectivity 2 is A327108.
Non-spanning edge-connectivity 2 is A327097.
Vertex-connectivity 3 is A327376.
Labeled graphs with vertex-connectivity 2 are A327198.
Set-systems with vertex-connectivity 2 are A327375.
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A000120, A013922, A048793, A070939, A259862, A326031, A326749, A327336.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]}]];
Select[Range[0,200],vertConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]

A327375 Number of set-systems with n vertices and vertex-connectivity 2.

Original entry on oeis.org

0, 0, 0, 72, 4752

Offset: 0

Gus Wiseman, Sep 05 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. 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.

BII-numbers for vertex-connectivity 2 are A327374.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
Labeled graphs with vertex-connectivity 2 are A327198.
The vertex-connectivity of the set-system with BII-number n is A327051(n).
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A013922, A259862, A322389, A322390, A323818, A327336.

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],{1,n}]],vertConnSys[Range[n],#]==2&]],{n,0,3}]

cp's OEIS Frontend

A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A327374 BII-numbers of set-systems with vertex-connectivity 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327375 Number of set-systems with n vertices and vertex-connectivity 2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica