Question: 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:

![](https://i.stack.imgur.com/8nY4Tm.png)

[(source)](http://www.math.illinois.edu/~stolee/blogimg/plane37.png)

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](http://hog.grinvin.org/) by searching using their drawing tool, but found nothing.

Answer: 
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:

![truncated cube](https://upload.wikimedia.org/wikipedia/commons/1/1d/Truncated_cubical_graph.neato.svg)
[Sage](http://sagemath.org/) 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](http://www.sagemath.org/doc/reference/graphs.html).