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 A327051 Vertex-connectivity of the set-system with BII-number 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 (list; graph; refs; listen; history; text; internal format)
 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 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 empty or duplicate edges) to obtain a non-connected set-system or singleton. Except for cointersecting set-systems (A326853), this is the same as cut-connectivity (A326786). LINKS Wikipedia, k-vertex-connected graph EXAMPLE Positions of first appearances of each integer, together with the corresponding set-systems, 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]-1||csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]] Table[vertConnSys[Union@@bpe/@bpe[n], bpe/@bpe[n]], {n, 0, 100}] CROSSREFS Cut-connectivity is A326786. Spanning edge-connectivity is A327144. Non-spanning edge-connectivity is A326787. The enumeration of labeled graphs by vertex-connectivity 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

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Last modified January 30 16:07 EST 2023. Contains 359945 sequences. (Running on oeis4.)