A326787 - OEIS

%I #10 Sep 26 2019 03:43:10

%S 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,

%T 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,

%U 2,3,3,2,2,1,1,1,1,1,1,1,2,3,1,1,3,4,2

%N Non-spanning edge-connectivity of the set-system with BII-number n.

%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 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.

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>

%e Positions of first appearances of each integer together with the corresponding set-systems:

%e      0: {}

%e      1: {{1}}

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

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

%e     85: {{1},{1,2},{1,3},{1,2,3}}

%e    341: {{1},{1,2},{1,3},{1,4},{1,2,3}}

%e   1365: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4}}

%e   5461: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,3,4}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t 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]]]]]]]]];

%t eConn[sys_]:=Length[sys]-Max@@Length/@Select[Subsets[sys],Length[csm[#]]!=1&];

%t Table[eConn[bpe/@bpe[n]],{n,0,100}]

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

%K nonn

%O 0,6

%A _Gus Wiseman_, Jul 25 2019