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

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..61.

Example

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}}

Mathematica

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&]

Crossrefs

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.

Sequence in context: A143723 A252713 A249404 * A327109 A327108 A295156

Adjacent sequences: A327371 A327372 A327373 * A327375 A327376 A327377

Keyword

nonn

Author

Gus Wiseman, Sep 04 2019

Status

approved