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 A327374 BII-numbers of set-systems with vertex-connectivity 2. 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified May 13 23:41 EDT 2021. Contains 343868 sequences. (Running on oeis4.)