A327097 BII-numbers of set-systems with non-spanning edge-connectivity 2.

5, 6, 17, 20, 24, 34, 36, 40, 48, 53, 54, 55, 60, 61, 62, 63, 65, 66, 68, 71, 72, 80, 86, 87, 89, 92, 93, 94, 95, 96, 101, 103, 106, 108, 109, 110, 111, 113, 114, 115, 121, 122, 123, 257, 260, 272, 308, 309, 310, 311, 316, 317, 318, 319, 320, 326, 327, 342

Offset: 1

Author

Gus Wiseman, Aug 20 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 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 non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any isolated vertices) to result in a disconnected or empty set-system.

Examples

			The sequence of all set-systems with non-spanning edge-connectivity 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  24: {{3},{1,3}}
  34: {{2},{2,3}}
  36: {{1,2},{2,3}}
  40: {{3},{2,3}}
  48: {{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}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
		

Crossrefs

Positions of 2's in A326787.

BII-numbers for vertex-connectivity 2 are A327082.

BII-numbers for non-spanning edge-connectivity 1 are A327099.

BII-numbers for non-spanning edge-connectivity > 1 are A327102.

BII-numbers for spanning edge-connectivity 2 are A327108.

Cf. A007146, A048793, A052446, A059166, A070939, A095983, A263296, A322335, A322338, A322395, A326031, A327041, A327069, A327111.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]==2&]