A327079 Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).

0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080

Offset: 0

Author

Gus Wiseman, Aug 25 2019

Keywords

• nonn

Comments

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Graphs with no bridges are counted by A095983 (2-edge-connected graphs).

Also labeled simple connected graphs covering n vertices with non-spanning edge-connectivity 1, where the non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

Crossrefs

Column k = 1 of A327149.

The non-covering version is A327231.

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

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

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

Cf. A001187, A006129, A052446, A059166, A322395, A327072, A327073, A327148.

Programs

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]]]]]]]]];
eConn[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}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,4}]

Formula

a(n) = A001187(n) - A322395(n) for n > 2. - Andrew Howroyd, Aug 27 2019

Inverse binomial transform of A327231.

Extensions

Terms a(6) and beyond from Andrew Howroyd, Aug 27 2019