A327376 BII-numbers of set-systems with vertex-connectivity 3.

2868, 2869, 2870, 2871, 2876, 2877, 2878, 2879, 2880, 2881, 2882, 2883, 2884, 2885, 2886, 2887, 2888, 2889, 2890, 2891, 2892, 2893, 2894, 2895, 2896, 2897, 2898, 2899, 2900, 2901, 2902, 2903, 2904, 2905, 2906, 2907, 2908, 2909, 2910, 2911, 2912, 2913, 2914

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..43.

Example

The sequence of all set-systems with vertex-connectivity 3 together with their BII-numbers begins:
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2869: {{1},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2870: {{2},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2871: {{1},{2},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2876: {{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2877: {{1},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2878: {{2},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2879: {{1},{2},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}
  2881: {{1},{1,2,3},{1,4},{2,4},{3,4}}
  2882: {{2},{1,2,3},{1,4},{2,4},{3,4}}
  2883: {{1},{2},{1,2,3},{1,4},{2,4},{3,4}}
  2884: {{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2885: {{1},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2886: {{2},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2887: {{1},{2},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2888: {{3},{1,2,3},{1,4},{2,4},{3,4}}
  2889: {{1},{3},{1,2,3},{1,4},{2,4},{3,4}}
  2890: {{2},{3},{1,2,3},{1,4},{2,4},{3,4}}
  2891: {{1},{2},{3},{1,2,3},{1,4},{2,4},{3,4}}

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, 3000], vertConnSys[Union@@bpe/@bpe[#], bpe/@bpe[#]]==3&]

Crossrefs

Positions of 3's in A327051.

BII-numbers for vertex-connectivity 2 are A327374.

BII-numbers for spanning edge-connectivity >= 3 are A327110.

The enumeration of labeled graphs by vertex-connectivity is A327334.

Cf. A000120, A013922, A048793, A070939, A259862, A323818, A326031, A326753, A326786, A327375.

Sequence in context: A286007 A235960 A235571 * A254587 A254580 A254355

Adjacent sequences: A327373 A327374 A327375 * A327377 A327378 A327379

Keyword

nonn

Author

Gus Wiseman, Sep 05 2019

Status

approved