A327147 -id:A327147 - cp's OEIS frontend

A263296 Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.

1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 10, 8, 2, 1, 44, 52, 41, 15, 3, 1, 191, 351, 352, 121, 25, 3, 1, 1229, 3714, 4820, 2159, 378, 41, 4, 1, 13588, 63638, 113256, 68715, 14306, 1095, 65, 4, 1, 288597, 1912203, 4602039, 3952378, 1141575, 104829, 3441, 100, 5, 1

Offset: 1

Christian Stump, Oct 13 2015

This is spanning edge-connectivity. The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. The non-spanning edge-connectivity of a graph (A327236) is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty. Compare to vertex-connectivity (A259862). - Gus Wiseman, Sep 03 2019

			Triangle begins:
     1;
     1,    1;
     2,    1,    1;
     5,    3,    2,    1;
    13,   10,    8,    2,   1;
    44,   52,   41,   15,   3,  1;
   191,  351,  352,  121,  25,  3, 1;
  1229, 3714, 4820, 2159, 378, 41, 4, 1;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..78 (rows 1..12)
FindStat - Combinatorial Statistic Finder, The edge connectivity of a graph.
Jens M. Schmidt, Combinatorial Data
Gus Wiseman, Unlabeled graphs with 5 vertices organized by spanning edge-connectivity (isolated vertices not shown).

Row sums give A000088, n >= 1.
Columns k=0..10 are A000719, A052446, A052447, A052448, A241703, A241704, A241705, A324096, A324097, A324098, A324099.
Number of graphs with edge connectivity at least k for k=1..10 are A001349, A007146, A324226, A324227, A324228, A324229, A324230, A324231, A324232, A324233.
The labeled version is A327069.
Cf. A002494, A095983, A259862, A327076, A327108, A327109, A327111, A327144, A327145, A327147, A327236.

a(22)-a(55) added by Andrew Howroyd, Aug 11 2019

A327144 Spanning edge-connectivity of the set-system with BII-number n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 31 2019

nonn

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.

			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}}

Dominated by A327103.
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 1's are A327111.
Positions of 2's are A327108.
Positions of first appearance of each integer are A327147.
Cf. A000120, A048793, A070939, A322338, A323818, A326031, A327041, A327069, A327076, A327130, A327145.

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}]

A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

Original entry on oeis.org

116, 117, 118, 119, 124, 125, 126, 127, 1796, 1797, 1798, 1799, 1904, 1905, 1906, 1907, 1908, 1909, 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919, 1924, 1925, 1926, 1927, 2032, 2033, 2034, 2035, 2036, 2037, 2038, 2039, 2040, 2041, 2042, 2043, 2044

Offset: 1

Gus Wiseman, Oct 03 2019

nonn

			The sequence of all set-systems with spanning edge-connectivity 3 together with their BII-numbers begins:
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   117: {{1},{1,2},{1,3},{2,3},{1,2,3}}
   118: {{2},{1,2},{1,3},{2,3},{1,2,3}}
   119: {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
   124: {{1,2},{3},{1,3},{2,3},{1,2,3}}
   125: {{1},{1,2},{3},{1,3},{2,3},{1,2,3}}
   126: {{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1797: {{1},{1,2},{1,4},{2,4},{1,2,4}}
  1798: {{2},{1,2},{1,4},{2,4},{1,2,4}}
  1799: {{1},{2},{1,2},{1,4},{2,4},{1,2,4}}
  1904: {{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1905: {{1},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1906: {{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1907: {{1},{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1908: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1909: {{1},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1910: {{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1911: {{1},{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}

Positions of 3's in A327144.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A001187, A095983, A322395, A326749, A326753, A326787, A327069, A327071, A327103, A327130, A327146, A327147.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==3&]

A327234 Smallest BII-number of a set-system with cut-connectivity n.

Original entry on oeis.org

0, 1, 4, 52, 2868

Offset: 0

Gus Wiseman, Sep 03 2019

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.
We define the cut-connectivity (A326786) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).
Conjecture: a(n > 1) = A327373(n) = the BII-number of K_n.

			The sequence of terms together with their corresponding set-systems:
     0: {}
     1: {{1}}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

The same for spanning edge-connectivity is A327147.
The cut-connectivity of the set-system with BII-number n is A326786(n).
Cf. A000120, A002450, A029931, A048793, A070939, A259862, A326031, A327082, A327098, A327125, A327126, A327127, A327373.

cp's OEIS Frontend

A263296 Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.

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A327144 Spanning edge-connectivity of the set-system with BII-number n.

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A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

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A327234 Smallest BII-number of a set-system with cut-connectivity n.

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