A194088
The number of biconnected squaregraphs that contain n squares.
Original entry on oeis.org
1, 1, 2, 6, 18, 72, 318, 1601, 8417
Offset: 1
For n=5 the a(5)=18 solutions are the 12 pentominoes, plus the "5-cogwheel", plus five others obtained by "tearing" the P-pentomino or the 5-cogwheel apart at one edge.
- V. Chepoi, F. Dragan, Y. Vaxès, Center and diameter problem in planar quadrangulations and triangulations, SODA 13 (2002), 346--355.
- H.-J. Bandelt, V.Chepoi, and D. Eppstein, Combinatorics and geometry of finite and infinite squaregraphs, SIAM Journal on Discrete Mathematics 24 (2010), 1399--1440.
A194089
The number of labeled biconnected squaregraphs that contain n squares.
Original entry on oeis.org
1, 3, 12, 57, 295, 1615, 9190, 53811, 322078
Offset: 1
A biconnected squaregraph is "labeled" if one of the edges on its periphery is given an orientation. The a(2)=3 examples with n=2 are the domino together with two possible ways to orient one of the edges on its "long" side, and one way to orient an edge on the short side. If a squaregraph has m automorphisms and perimeter 2p, the number of different ways to label it is 4p/m.
A194090
The number of biconnected squaregraphs of perimeter 2*n.
Original entry on oeis.org
1, 1, 1, 3, 8, 35, 172, 1121, 8017, 63213
Offset: 1
For n=4 the a(4)=3 cases of perimeter 8 are the two trominoes and the square tetromino.
A194092
The number of biconnected squaregraphs with n vertices.
Original entry on oeis.org
1, 1, 0, 1, 0, 1, 0, 2, 1, 5, 2, 17, 11, 65, 54, 299, 333, 1508, 2041, 8215, 12971
Offset: 1
For n=11 the a(11)=2 solutions are the P pentomino and the 5-cogwheel.
The cases n=1 and n=2 are somewhat controversial, depending on whether the graphs K_1 and K_2 are considered to be "biconnected".
A194093
The number of labeled biconnected squaregraphs with n vertices.
Original entry on oeis.org
1, 1, 0, 1, 0, 3, 0, 12, 2, 55, 22, 278, 182, 1506, 1378, 8595, 10040, 51134, 71814, 314693, 509200
Offset: 1
[See A194089 for the definition of "labeling".] For n=11 the P-pentomino can be labeled in 20 ways; the 5-cogwheel in only 2; hence a(11)=22.
The values of a(1) and a(2) depend on whether or not one believes that
the small complete graphs K_1 and K_2 respectively are biconnected.
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