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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

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

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Last modified April 21 12:59 EDT 2021. Contains 343153 sequences. (Running on oeis4.)