proposed

approved

editing

proposed

Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108, A327148.

proposed

editing

editing

proposed

These are Also positions of terms >=2 in A326787.

The sequence of all set-systems with non-spanning 2-edge-connectivity >= 2 together with their BII-numbers begins:

A set-system has non-spanning 2-edge-connectivity >= 2 if 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).

A set-system has non-spanning 2-edge-connectivity >= 2 if 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).

A set-system has non-spanning 2-edge-connectivity >= 2 if 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).

allocated for Gus WisemanBII-numbers of set-systems with non-spanning edge-connectivity >= 2.

5, 6, 17, 20, 21, 24, 34, 36, 38, 40, 48, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121

1,1

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.

A set-system has non-spanning 2-edge-connectivity >= 2 if 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) are all endpoints (edges intersecting only one other edge).

The sequence of all set-systems with non-spanning 2-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}}

21: {{1},{1,2},{1,3}}

24: {{3},{1,3}}

34: {{2},{2,3}}

36: {{1,2},{2,3}}

38: {{2},{1,2},{2,3}}

40: {{3},{2,3}}

48: {{1,3},{2,3}}

52: {{1,2},{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}}

56: {{3},{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}}

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&]

Graphs with spanning edge-connectivity >= 2 are counted by A095983.

Graphs with non-spanning edge-connectivity >= 2 are counted by A322395.

These are positions of terms >=2 in A326787.

BII-numbers for non-spanning edge-connectivity 2 are A327097.

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

BII-numbers for spanning edge-connectivity >= 2 are A327109.

Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108.

allocated

nonn

Gus Wiseman, Aug 23 2019

approved

editing