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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 February 27 12:18 EST 2020. Contains 332305 sequences. (Running on oeis4.)