Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/Pseudotree.html">Pseudotree</a>

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Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024

Also unlabeled connected graphs covering n vertices with at most n edges. Also unlabeled connected choosable graphs covering n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. For these definitions this definition we have a(1) = 0. - Gus Wiseman, Feb 20 2024

The Representatives of the a(0) = 1 through a(5) = 8 graph edge setsgraphs:

Also unlabeled connected graphs with covering n vertices, none isolated, and with at most n edges. Also unlabeled connected choosable graphs with covering n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. For these definitions we have a(1) = 0. - Gus Wiseman, Feb 20 2024

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]]]]]]]]];

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];

Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]<=1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]], {n, 0, 5}] (* Gus Wiseman, Feb 20 2024 *)

Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, `A367868, ~A367916, A368951, `A369194, A369197, A370317, A370318.

Cf. A000055, A000081, A001429, (labeled A057500), A134964 (number of pseudoforests, labeled A133686).

For any number of edges we have A001187, unlabeled A001349.

The case of equality is A001429, labeled A057500 (covering A370317, counting only covered vertices A370318).

The non-connected non-covering version complement is A134964, A140636, labeled A133686A140638.

The connected complement is A140636, labeled A140638, covering , non-connected .

The nonNon-connected version is : A368834 (labeled A367869) or A370316 (labeled A369191).

The version with loops is new, labeled A369197 (non-connected A369194).

A006125 A001187 counts connected graphs, A000088 unlabeled A001349.

A006129 A006125 counts covering simple graphs, A002494 unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

Cf. A000272, A006649, A116508, A143543 lab_tri_gr, A367862, A367863, A367868 labgra_cov_contra_aoc, A367916 setsys_cov_vts_eq_eds, A367917 bii_vts_eq_eds, A368951.

Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, `A367868, ~A367916, A368951, `A369194, A369197, A370317, A370318.

Also unlabeled connected graphs with n vertices, none isolated, and at most n edges. Also unlabeled connected choosable graphs with n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. For these definitions we have a(1) = 0. - Gus Wiseman, Feb 20 2024

From Gus Wiseman, Feb 20 2024: (Start)

The a(0) = 1 through a(5) = 8 graph edge sets:

{} . {12} {12,13} {12,13,14} {12,13,14,15}

{12,13,23} {12,13,24} {12,13,14,25}

{12,13,14,23} {12,13,24,35}

{12,13,24,34} {12,13,14,15,23}

{12,13,14,23,25}

{12,13,14,23,45}

{12,13,14,25,35}

{12,13,24,35,45}

(End)

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]]]]]]]]];

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];

Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]<=1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]], {n, 0, 5}] (* Gus Wiseman, Feb 20 2024 *)

For any number of edges we have A001187, unlabeled A001349.

The case of equality is A001429, labeled A057500 (covering A370317, counting only covered vertices A370318).

The labeled version is A129271.

The non-connected non-covering version is A134964, labeled A133686.

The connected complement is A140636, labeled A140638, covering , non-connected .

The non-connected version is A368834 (labeled A367869) or A370316 (labeled A369191).

The version with loops is new, labeled A369197 (non-connected A369194).

A006125 counts graphs, A000088 unlabeled.

A006129 counts covering graphs, A002494 unlabeled.

A062734 counts connected graphs by number of edges.

Cf. A000272, A006649, A116508, A143543 lab_tri_gr, A367862, A367863, A367868 labgra_cov_contra_aoc, A367916 setsys_cov_vts_eq_eds, A367917 bii_vts_eq_eds, A368951.

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