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A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1). 8
0, 0, 1, 3, 18, 250, 5475, 191541, 11065572, 1104254964, 201167132805, 69828691941415, 47150542741904118, 62354150876493659118, 161919876753750972738791, 827272271567137357352991705, 8331016130913639432634637862600, 165634930763383717802534343776893928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A bridge is an edge whose removal disconnected the graph, while an endpoint is a vertex belonging to only one edge. 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.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50

FORMULA

Binomial transform of A327079.

EXAMPLE

The a(2) = 1 through a(4) = 18 edge-sets:

  {12}  {12}  {12}

        {13}  {13}

        {23}  {14}

              {23}

              {24}

              {34}

              {12,13,24}

              {12,13,34}

              {12,14,23}

              {12,14,34}

              {12,23,34}

              {12,24,34}

              {13,14,23}

              {13,14,24}

              {13,23,24}

              {13,24,34}

              {14,23,24}

              {14,23,34}

MATHEMATICA

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]]]]]]]]];

edgeConnSys[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];

Table[Length[Select[Subsets[Subsets[Range[n], {2}]], edgeConnSys[#]==1&]], {n, 0, 4}]

CROSSREFS

Column k = 1 of A327148.

The covering version is A327079.

Connected bridged graphs (spanning edge-connectivity 1) are A327071.

BII-numbers of set-systems with non-spanning edge-connectivity 1 are A327099.

Covering set-systems with non-spanning edge-connectivity 1 are A327129.

Cf. A001187, A052446, A322395, A327072, A327073, A327102.

Sequence in context: A159640 A038061 A232916 * A279490 A299431 A222790

Adjacent sequences:  A327228 A327229 A327230 * A327232 A327233 A327234

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 01 2019

EXTENSIONS

Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019

STATUS

approved

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Last modified July 10 00:25 EDT 2020. Contains 335570 sequences. (Running on oeis4.)