A326787 Non-spanning edge-connectivity of the set-system with BII-number n.

0, 1, 1, 0, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 3, 1, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 3, 1, 2, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 4, 2

Offset

0,6

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 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 non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

Links

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

Wikipedia, k-edge-connected graph

Example

Positions of first appearances of each integer together with the corresponding set-systems:
     0: {}
     1: {{1}}
     5: {{1},{1,2}}
    21: {{1},{1,2},{1,3}}
    85: {{1},{1,2},{1,3},{1,2,3}}
   341: {{1},{1,2},{1,3},{1,4},{1,2,3}}
  1365: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4}}
  5461: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,3,4}}

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]]]]]]]]];
eConn[sys_]:=Length[sys]-Max@@Length/@Select[Subsets[sys], Length[csm[#]]!=1&];
Table[eConn[bpe/@bpe[n]], {n, 0, 100}]

Crossrefs

Cf. A000120, A013922, A048793, A070939, A095983, A322336, A322338 (same for MM-numbers), A326031, A326749, A326753, A326786 (vertex-connectivity).

Sequence in context: A356818 A265863 A083747 * A246271 A049334 A054924

Adjacent sequences: A326784 A326785 A326786 * A326788 A326789 A326790

Keyword

nonn

Author

Gus Wiseman, Jul 25 2019

Status

approved