A187927 -id:A187927 - cp's OEIS frontend

A187928 Number of embeddings on the sphere of planar graphs with n edges having connectivity exactly 2 and minimum vertex degree at least 3.

1, 2, 4, 15, 42, 135, 440, 1480, 5106, 17890, 63264, 226018, 812354, 2936837, 10666188, 38901190, 142386358

Offset: 10

Stuart E Anderson, Mar 16 2011

The graphs are 2-connected, but not 3-connected. The graphs were enumerated using plantri (by B.D. McKay & G. Brinkmann) for the purpose of finding compound perfect squared squares. If all graphs with n edges are generated then all compound squares in order n-1 can be obtained from them. Graphs with minimum degree at least 3 are also called homeomorphically irreducible.

Stuart Anderson, Compound Perfect Squared Squares
B. D. McKay and G. Brinkmann, Plantri Guide
Gunnar Brinkmann and Brendan McKay, Guide to using plantri [Cached copy, with permission]

Antidiagonal sums of A378077.

Cf. A034889, A181340, A187927, A021103, A378076.

plantri
```
plantri -p -c2 -m3 -e# -x -u -v n
```

plantri -pc2m3e#xuv n # to count graphs by node number (n) and edge number (#)

a(22) corrected by Stuart E Anderson, Feb 24 2013
a(23)-a(26) from Lorenz Milla, Oct 08 2013
a(11) corrected by Andrew Howroyd, Nov 15 2024

A378077 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of planar graphs with n vertices and k faces having connectivity exactly 2 and minimum vertex degree at least 3, k=6..2n-5.

Original entry on oeis.org

1, 1, 1, 3, 7, 2, 1, 8, 35, 60, 47, 12, 0, 5, 72, 307, 647, 652, 325, 59, 0, 3, 86, 776, 3395, 7647, 9582, 6654, 2442, 368, 0, 0, 45, 1041, 9091, 38876, 94278, 136628, 121204, 64232, 18916, 2363, 0, 0, 18, 827, 14407, 111076, 468211, 1192511, 1937266, 2049784, 1409199, 607746, 150161, 16253

Offset: 6

Andrew Howroyd, Nov 15 2024

The graphs are 2-connected, but not 3-connected. Graphs with minimum degree at least 3 are also called homeomorphically irreducible.

The number of edges is n + k - 2.

			Triangle begins:
  n\k| 6  7   8     9    10     11     12      13      14     15     16    17
-----+------------------------------------------------------------------------
   6 | 1, 1;
   7 | 1, 3,  7,    2;
   8 | 1, 8, 35,   60,   47,    12;
   9 | 0, 5, 72,  307,  647,   652,   325,     59;
  10 | 0, 3, 86,  776, 3395,  7647,  9582,   6654,   2442,   368;
  11 | 0, 0, 45, 1041, 9091, 38876, 94278, 136628, 121204, 64232, 18916, 2363;
  ...

Andrew Howroyd, Table of n, a(n) for n = 6..95 (rows 6..14)

Rows sums are A187927.
Antidiagonals sums give A187928.
Cf. A378075.

T(n,k) = A212438(n,k) - A378075(n,k).

A378074 Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 1, 2, 9, 47, 420, 4673, 63253, 927238, 14342093, 229607392, 3776227106, 63482545872, 1087322656758, 18927037827561

Offset: 1

Andrew Howroyd, Nov 15 2024

Homeomorphically irreducible means each vertex has a degree of at least 3.

Row sums of A378075.

Cf. A000944, A034889, A187927, A378076.

a(n) = A000944(n) + A187927(n).

cp's OEIS Frontend

A187928 Number of embeddings on the sphere of planar graphs with n edges having connectivity exactly 2 and minimum vertex degree at least 3.

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A378077 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of planar graphs with n vertices and k faces having connectivity exactly 2 and minimum vertex degree at least 3, k=6..2n-5.

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A378074 Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n nodes.

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