

A327051


Vertexconnectivity of the setsystem with BIInumber n.


16



0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
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OFFSET

0,53


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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
The vertexconnectivity of a setsystem is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a nonconnected setsystem or singleton. Except for cointersecting setsystems (A326853), this is the same as cutconnectivity (A326786).


LINKS

Table of n, a(n) for n=0..86.
Wikipedia, kvertexconnected graph


EXAMPLE

Positions of first appearances of each integer, together with the corresponding setsystems, are:
0: {}
4: {{1,2}}
52: {{1,2},{1,3},{2,3}}
2868: {{1,2},{1,3},{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]1csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]
Table[vertConnSys[Union@@bpe/@bpe[n], bpe/@bpe[n]], {n, 0, 100}]


CROSSREFS

Cutconnectivity is A326786.
Spanning edgeconnectivity is A327144.
Nonspanning edgeconnectivity is A326787.
The enumeration of labeled graphs by vertexconnectivity is A327334.
Cf. A000120, A013922, A029931, A048793, A070939, A259862, A322389, A323818, A326031, A327125, A327198, A327336.
Sequence in context: A258383 A037805 A327144 * A275301 A282542 A271518
Adjacent sequences: A327048 A327049 A327050 * A327052 A327053 A327054


KEYWORD

nonn


AUTHOR

Gus Wiseman, Sep 02 2019


STATUS

approved



