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A327376
BII-numbers of set-systems with vertex-connectivity 3.
1
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.
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.
Sequence in context: A286007 A235960 A235571 * A254587 A254580 A254355
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 05 2019
STATUS
approved