A002218 -id:A002218 - cp's OEIS frontend

A034889 Number of embeddings on the sphere of 2-connected planar graphs with n nodes.

1, 3, 10, 61, 564, 7593, 123874, 2262877, 44190279, 904777809, 19207129217, 419870351012, 9405626692325

Offset: 3

Ronald C. Read

The complete graph on two vertices is sometimes considered to be 2-connected (or nonseparable). Compare A002218 with A021103. - Andrew Howroyd, Mar 01 2021

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem, 58 (2007), 323-357. [Expanded version]
Eric Weisstein's World of Mathematics, Planar Embedding.

Row sums of A342060.

Cf. A002218, A005142, A021103.

a(8)-a(15) added by Mohammadreza Jooyandeh, Sep 03 2013

A241767 Number of simple connected graphs with n nodes and exactly 1 articulation point (cutpoints).

Original entry on oeis.org

0, 0, 1, 2, 7, 33, 244, 2792, 52448, 1690206, 96288815, 9873721048, 1841360945834, 629414405238720, 397024508142598996, 464923623652122023478, 1016016289424631486429082, 4162473006943138723685574978, 32096861904411547975392065322659

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

nonn

Terms may be computed from A004115. See formula. There is an obvious bijection between a connected graph with 1 articulation point and a multiset of at least two rooted nonseparable graphs joined at the root node. - Andrew Howroyd, Nov 24 2020

Andrew Howroyd, Table of n, a(n) for n = 1..26
Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=1 of A325111.
Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.
Cf. A004115 (rooted and without articulation points).

G.f.: x/(Product_{k>=1} (1 - x^k)^A004115(k+1)) - x - Sum_{k>=1} A004115(k)*x^k. - Andrew Howroyd, Nov 24 2020

Terms a(11) and beyond from Andrew Howroyd, Nov 24 2020

A325111 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k articulation vertices, (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 3, 2, 1, 0, 0, 10, 7, 3, 1, 0, 0, 56, 33, 17, 5, 1, 0, 0, 468, 244, 101, 32, 7, 1, 0, 0, 7123, 2792, 890, 242, 60, 9, 1, 0, 0, 194066, 52448, 11468, 2461, 527, 97, 12, 1, 0, 0, 9743542, 1690206, 239728, 35839, 6056, 1029, 155, 15, 1, 0, 0

Offset: 0

Andrew Howroyd, Sep 05 2019

Articulation vertices are also called cutpoints. These are vertices that when removed increase the component count of the graph.

			Triangle begins:
     1;
     1     0;
     1,    0,   0;
     1,    1,   0,   0;
     3,    2,   1,   0,  0;
    10,    7,   3,   1,  0, 0;
    56,   33,  17,   5,  1, 0, 0;
   468,  244, 101,  32,  7, 1, 0, 0;
  7123, 2792, 890, 242, 60, 9, 1, 0, 0;
  ...

Eric Weisstein's World of Mathematics, Articulation Vertex

Columns k=0..5 are A002218(n>1), A241767, A241768, A241769, A241770, A241771.
Row sums are A001349.
Cf. A327077, A370064 (labeled version).

Diagonal for k = n inserted by Andrew Howroyd, Feb 25 2024

A241769 Number of simple connected graphs with n nodes and exactly 3 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 32, 242, 2461, 35839

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=3 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241770 Number of simple connected graphs with n nodes and exactly 4 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 60, 527, 6056

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=4 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A241771 Number of simple connected graphs with n nodes and exactly 5 articulation points (cutpoints).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 9, 97, 1029

Offset: 1

Travis Hoppe and Anna Petrone, Apr 28 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Articulation Vertex

Column k=5 of A325111.

Cf. other simple connected graph sequences with k articulation points A002218, A241767, A241768, A241769, A241770, A241771.

A322117 Number of non-isomorphic blobs (2-connected weak antichains) of multisets of weight n.

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 21, 27, 79, 185, 554

Offset: 0

Gus Wiseman, Nov 26 2018

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 21 blobs:
  (1)  (11)    (111)      (1111)        (11111)          (111111)
       (12)    (122)      (1122)        (11222)          (111222)
       (1)(1)  (123)      (1222)        (12222)          (112222)
               (1)(1)(1)  (1233)        (12233)          (112233)
                          (1234)        (12333)          (122222)
                          (11)(11)      (12344)          (122333)
                          (12)(12)      (12345)          (123333)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)  (123344)
                                                         (123444)
                                                         (123455)
                                                         (123456)
                                                         (111)(111)
                                                         (112)(122)
                                                         (122)(122)
                                                         (123)(123)
                                                         (123)(233)
                                                         (134)(234)
                                                         (11)(11)(11)
                                                         (12)(12)(12)
                                                         (12)(13)(23)
                                                         (1)(1)(1)(1)(1)(1)

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A002218, A007718, A013922, A275307, A286520, A293994, A304118, A304887, A319719, A319721, A322110, A322118.

A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= k <= n*(n-1)/2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1, 0, 1, 4, 20, 50, 82, 94, 81, 59, 38, 20, 10, 5, 2, 1, 1, 0, 1, 6, 40, 161, 429, 780, 1076, 1197, 1114, 885, 622, 386, 215, 112, 55, 24, 11, 5, 2, 1, 1, 0, 1, 7, 70, 433, 1729, 4796

Offset: 1

Andrew Howroyd, Nov 23 2020

			Triangle T(n,k) begins:
======================================================
n/k | 0  1  2  3  4  5  6  7  8   9  10 11 12 13 14 15
----+-------------------------------------------------
  1 | 0;
  2 |    1;
  3 |       0, 1;
  4 |          0, 1, 1, 1;
  5 |             0, 1, 2, 3, 2,  1,  1;
  6 |                0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..576 (rows 1..16, extracted from Robinson's tables)
R. W. Robinson, Tables of 2-Connected and 3-Connected Graphs by Nodes and Edges, Table IV, pages 4-9.
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]

Row sums are A002218.
Column sums are A010355.
Cf. A054923, A054924, A123534, A339070 (transpose), A339072.

A006289 Number of series-reduced 2-connected graphs with n nodes.

Original entry on oeis.org

1, 3, 19, 149, 2581, 84151, 5201856, 577050233, 113372069299, 39618015318982, 24916462761069296, 28563626972509456884, 60366734349116636660402, 237406975840304068884168139, 1750330441810569047176394509086

Offset: 4

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 4..25
R. W. Robinson, Tables
R. W. Robinson, Tables [Local copy, with permission]
R. W. Robinson and T. R. S. Walsh, Inversion of cycle index sum relations for 2- and 3-connected graphs, J. Combin. Theory Ser. B. 57 (1993), 289-308. See also.
T. R. S. Walsh, Counting unlabeled three-connected and homeomorphically irreducible two-connected graphs, J. Combin. Theory Ser. B 32 (1982), no. 1, 12-32.

Row sums of A339069.

Cf. A006290, A339068.

Robinson and Walsh list first 25 terms.

A010357 Number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 32, 90, 279, 942, 3468, 13777, 57747, 254671, 1170565, 5580706, 27487418, 139477796, 727458338, 3893078684, 21346838204, 119787629215, 687200870250

Offset: 1

N. J. A. Sloane

Original name: Multi-edge stars with n edges.

			From _Andrew Howroyd_, Nov 23 2020: (Start)
The a(1) = 1 graph is a single edge (K_2 = P_2).
The a(2) = 1 graph is a double edge.
The a(3) = 2 graphs are a triple edge and the triangle (K_3).
The a(4) = 3 graphs are a quadruple edge, a triangle with one double edge and the square (C_4).
(End)

George A. Baker Jr. and John M. Kincaid, The continuous-spin Ising model, g0:phi4:d field theory, and the renormalization group, J. Statist. Phys. 24 (1981), no. 3, 469-528.
Brendan McKay and Adolfo Piperno, nauty and Traces, programs for computing automorphism groups of graphs and digraphs.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 1 through a(6) = 14 unlabeled 2-connected multigraphs.

Row sums of A339160.
Cf. A050535, A076864, A010355, A010359.
A002218 counts unlabeled 2-connected graphs.
A013922 counts labeled 2-connected graphs.
A322140 is a labeled version.
Cf. A002905, A006444, A007718, A275307, A304887.

Name changed by Andrew Howroyd, Dec 05 2020
a(11)-a(20) added using geng/multig from nauty by Andrew Howroyd, Dec 05 2020
a(21)-a(23) from Sean A. Irvine, Apr 18 2024

cp's OEIS Frontend

A034889 Number of embeddings on the sphere of 2-connected planar graphs with n nodes.

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A241767 Number of simple connected graphs with n nodes and exactly 1 articulation point (cutpoints).

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A325111 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k articulation vertices, (0 <= k <= n).

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A241769 Number of simple connected graphs with n nodes and exactly 3 articulation points (cutpoints).

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A241770 Number of simple connected graphs with n nodes and exactly 4 articulation points (cutpoints).

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A241771 Number of simple connected graphs with n nodes and exactly 5 articulation points (cutpoints).

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A322117 Number of non-isomorphic blobs (2-connected weak antichains) of multisets of weight n.

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A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= k <= n*(n-1)/2).

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A006289 Number of series-reduced 2-connected graphs with n nodes.

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A010357 Number of unlabeled nonseparable (or 2-connected) loopless multigraphs with n edges.

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