Table of planar connected graphs For the other day I need to use the table of planar connected graph with few vertex. In the wolfram's mathworld , they listed only the graph with $4$ vertex.
Does anyone knows webpages or pdf on the web which carries the table of planar connected graph with up to $10$ edges?
 A: You can generate all connected graphs with no more than 10 edges quickly using Brendan McKay's NAUTY program availible at http://pallini.di.uniroma1.it/ using (example for 11 vertices, which is maximum):

geng -c 11 0:10

and filter then using "planarg" from the same package. That is if you really mean "up to 10 edges" (not vertices). I've posted a list of all such 3386 such graphs at http://pastebin.com/P5vrZH7X (obtained in the way just described, in graph6 format).
If it is the planar connected graphs with up to 10 vertices you want you may find them at House of Graphs: http://hog.grinvin.org/Planar
A: Not quite an answer, but two possible sources:
(1) A003094 lists the number of unlabeled connected planar simple graphs with $n$ nodes:
$$1, 1, 1, 2, 6, 20, 99, 646, 5974, 71885, 1052805, 17449299, 313372298\;.$$
(2) Read, Ronald C., and Robin J. Wilson. An atlas of graphs. Clarendon Press, 1998.
I can't access more than the table of contents of this book at the moment:



A: In Sage, see www.sagemath.org, you find the algorithm
G. Brinkmann and B.D. McKay, Fast generation of planar graphs, MATCH-Communications in Mathematical and in Computer Chemistry, 58(2):323-357, 2007.
implemented, for which the optional package plantri needs to be installed. From the documentation:
    An iterator over connected planar graphs using the plantri generator.

    This uses the plantri generator (see [plantri]_) which is available
    through the optional package plantri.

    .. NOTE::

        The non-3-connected graphs will be returned several times, with all
        its possible embeddings.

Here are a few tests:
sage: %time len(list(graphs.planar_graphs(4)))
CPU times: user 3.23 ms, sys: 7.93 ms, total: 11.2 ms
Wall time: 21.6 ms
6

sage: %time len(list(graphs.planar_graphs(5)))
CPU times: user 11.9 ms, sys: 8.35 ms, total: 20.3 ms
Wall time: 31.9 ms
25

sage: %time len(list(graphs.planar_graphs(6)))
CPU times: user 72.5 ms, sys: 11.7 ms, total: 84.2 ms
Wall time: 96.3 ms
179

sage: %time len(list(graphs.planar_graphs(7)))
CPU times: user 920 ms, sys: 67.3 ms, total: 988 ms
Wall time: 1 s
2014

sage: %time len(list(graphs.planar_graphs(8)))
CPU times: user 19.6 s, sys: 914 ms, total: 20.5 s
Wall time: 20.6 s
31178

A: A lot depends on what the question means, in particular are the graphs simple, and is equivalence defined by embedding-preserving isomorphism or all graph isomorphisms.
For simple graphs up to general isomorphism, see my page for files up to 11 vertices. 
For simple graphs up to embedding-preserving isomorphism, use plantri to make them.  The command for $n$ vertices is "plantri -pc1m1 $n$", or "plantri -opc1m1 $n$" if mirror-image doesn't count as isomorphism. (At first, add "-u" to just count rather than output.) Tables up to 14 vertices are in this paper, Table 22.
A: Close to what you are looking for, there is


*

*An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces by David Jackson and Terry I. Visentin. (CRC press) (Google books)


This book includes tables of all of the topological maps of genus 0 with up to five edges, ordered by number of vertices (it includes many other tables as well).  Note that a topological map of genus 0 can be seen as a connected planar graph with a bit of extra structure (a cyclic ordering of half-edges around each vertex).
