# Does the cubic planar graph with 6 3-faces and 6 7-faces have a name?

There is exactly one cubic planar graph with six 3-faces and six 7-faces (and no other faces). Surely it must have a name. What is it called?

Here is a picture of the graph embedded on the plane with a point at infinity:

A slightly more general question: How can I find out what the names are of semi-famous graphs? A graph is semi-famous if it has an established name but is not easily found in a standard textbook on the subject.

I did try the House of Graphs by searching using their drawing tool, but found nothing.

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There are a few on Wikipedia: http://en.wikipedia.org/wiki/Category:Individual_graphs, and there are more on MathWorld: http://mathworld.wolfram.com/topics/SimpleGraphs.html. There is a page on Wikipedia with pictures: http://en.wikipedia.org/wiki/Gallery_of_named_graphs

Unfortunately I don't have an answer to your question, but I would call it a "partially truncated cube", where the non-truncated vertices are opposite. Accordingly, you could say that it is constructed from the truncated cube by performing a $\Delta$-$Y$ contraction on two maximally distant triangles. I'm moving this comment to an answer in order to attach a picture of the truncated cube:

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You beat me to "partially truncated cube" while I was off looking for a good wikipedia reference. –  Lee Mosher Mar 8 '13 at 16:53

Sage has search facilities for graphs with specified properties, and "knows" a large number of "named" graphs. As you work within a system running Python, you can do much more much easier, compared to what you get just by searching online databases using a www interface. The manual of the corresponding part is here.

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I'm familiar with the Sage graph library, but not familiar with a functionality that allows me to give it a graph and ask for its name. The only thing I can think about is trying all graph.NAME objects and testing isomorphism (and some are parameterized). –  Derrick Stolee Mar 9 '13 at 14:10
This is a bit of a hack, but you can do something like this: sage: import inspect sage: g=inspect.getmembers(graphs, predicate=inspect.isfunction) will give you a correspondence between graph names and functions to create these named graphs (not all of entries of g would correspond to actual named graphs, so you'd need to be a bit careful) –  Dima Pasechnik Mar 9 '13 at 16:10