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%I #11 Aug 23 2019 14:15:35
%S 5,6,17,20,21,24,34,36,38,40,48,52,53,54,55,56,60,61,62,63,65,66,68,
%T 69,70,71,72,80,81,84,85,86,87,88,89,92,93,94,95,96,98,100,101,102,
%U 103,104,106,108,109,110,111,112,113,114,115,116,117,118,119,120,121
%N BII-numbers of set-systems with non-spanning edge-connectivity >= 2.
%C 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.
%C A set-system has non-spanning 2-edge-connectivity >= 2 if it is connected and any single edge can be removed (along with any non-covered vertices) without making the set-system disconnected or empty. Alternatively, these are connected set-systems whose bridges (edges whose removal disconnects the set-system or leaves isolated vertices) are all endpoints (edges intersecting only one other edge).
%e The sequence of all set-systems with non-spanning edge-connectivity >= 2 together with their BII-numbers begins:
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 17: {{1},{1,3}}
%e 20: {{1,2},{1,3}}
%e 21: {{1},{1,2},{1,3}}
%e 24: {{3},{1,3}}
%e 34: {{2},{2,3}}
%e 36: {{1,2},{2,3}}
%e 38: {{2},{1,2},{2,3}}
%e 40: {{3},{2,3}}
%e 48: {{1,3},{2,3}}
%e 52: {{1,2},{1,3},{2,3}}
%e 53: {{1},{1,2},{1,3},{2,3}}
%e 54: {{2},{1,2},{1,3},{2,3}}
%e 55: {{1},{2},{1,2},{1,3},{2,3}}
%e 56: {{3},{1,3},{2,3}}
%e 60: {{1,2},{3},{1,3},{2,3}}
%e 61: {{1},{1,2},{3},{1,3},{2,3}}
%e 62: {{2},{1,2},{3},{1,3},{2,3}}
%e 63: {{1},{2},{1,2},{3},{1,3},{2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t 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]]]]]]]]];
%t edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
%t Select[Range[0,100],edgeConn[bpe[#]]>=2&]
%Y Graphs with spanning edge-connectivity >= 2 are counted by A095983.
%Y Graphs with non-spanning edge-connectivity >= 2 are counted by A322395.
%Y Also positions of terms >=2 in A326787.
%Y BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y BII-numbers for non-spanning edge-connectivity 1 are A327099.
%Y BII-numbers for spanning edge-connectivity >= 2 are A327109.
%Y Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108, A327148.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 23 2019
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