|
|
A327144
|
|
Spanning edge-connectivity of the set-system with BII-number n.
|
|
20
|
|
|
0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,53
|
|
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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.
|
|
LINKS
|
|
|
EXAMPLE
|
Positions of first appearances of each integer together with the corresponding set-systems:
0: {}
1: {{1}}
52: {{1,2},{1,3},{2,3}}
116: {{1,2},{1,3},{2,3},{1,2,3}}
3952: {{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4}}
8052: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4}}
|
|
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]]]]]]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&];
Table[spanEdgeConn[Union@@bpe/@bpe[n], bpe/@bpe[n]], {n, 0, 100}]
|
|
CROSSREFS
|
The same for cut-connectivity is A326786.
The same for non-spanning edge-connectivity is A326787.
The same for vertex-connectivity is A327051.
Positions of first appearance of each integer are A327147.
Cf. A000120, A048793, A070939, A322338, A323818, A326031, A327041, A327069, A327076, A327130, A327145.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|