OFFSET
0,5
COMMENTS
A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 12 graphs with exactly one bridge.
FORMULA
E.g.f.: (x + Sum_{k>=2} A095983(k)*x^k/(k-1)!)^2/2. - Andrew Howroyd, Aug 25 2019
MATHEMATICA
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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#, i]]<n||Length[csm[Delete[#, i]]]>1, {i, Length[#]}], True]==1&]], {n, 0, 5}]
PROG
(PARI) \\ See A095983.
seq(n)={my(p=x*deriv(log(sum(k=0, n-1, 2^binomial(k, 2) * x^k / k!) + O(x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))^2/2), -(n+1)) } \\ Andrew Howroyd, Dec 28 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2019
EXTENSIONS
Terms a(6) and beyond from Andrew Howroyd, Aug 25 2019
STATUS
approved