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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
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.
Sequence in context: A159640 A038061 A232916 * A279490 A299431 A222790
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 March 29 03:41 EDT 2024. Contains 371264 sequences. (Running on oeis4.)