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A327111 BII-numbers of set-systems with spanning edge-connectivity 1. 22

%I #6 Aug 26 2019 12:40:08

%S 1,2,4,5,6,7,8,16,17,20,21,22,23,24,25,28,29,30,31,32,34,36,37,38,39,

%T 40,42,44,45,46,47,48,49,50,51,56,57,58,59,64,65,66,67,68,69,70,71,72,

%U 73,74,75,76,77,78,79,80,81,82,83,88,89,90,91,96,97,98,99

%N BII-numbers of set-systems with spanning edge-connectivity 1.

%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 The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

%e The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:

%e 1: {{1}}

%e 2: {{2}}

%e 4: {{1,2}}

%e 5: {{1},{1,2}}

%e 6: {{2},{1,2}}

%e 7: {{1},{2},{1,2}}

%e 8: {{3}}

%e 16: {{1,3}}

%e 17: {{1},{1,3}}

%e 20: {{1,2},{1,3}}

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

%e 22: {{2},{1,2},{1,3}}

%e 23: {{1},{2},{1,2},{1,3}}

%e 24: {{3},{1,3}}

%e 25: {{1},{3},{1,3}}

%e 28: {{1,2},{3},{1,3}}

%e 29: {{1},{1,2},{3},{1,3}}

%e 30: {{2},{1,2},{3},{1,3}}

%e 31: {{1},{2},{1,2},{3},{1,3}}

%e 32: {{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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];

%t Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

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

%Y BII-numbers for vertex-connectivity 1 are A327098.

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

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

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

%Y Set-systems with spanning edge-connectivity 2 are counted by A327130.

%Y Graphs with spanning edge-connectivity 1 are counted by A327145.

%Y Graphs with spanning edge-connectivity 2 are counted by A327146.

%Y Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 25 2019

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)