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

Author

Gus Wiseman, Jul 25 2019

Keywords

• nonn

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.

Examples

			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}}
		

Links

• Wikipedia, k-edge-connected graph

Crossrefs

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

Programs

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}]