How to get a list of 3-connected graphs?

Question

I'd like to use GraphData to get a list of graphs with vertex connectivity at least 3. I know GraphData["Nonplanar", 6] gives all the nonplanar graphs on 6 vertices, but I can't figure out how to get GraphData to do a selection based on the "VertexConnectivity" property.

Answer 1

It's been suggested that I combine the helpful information given in comments into an answer. To deal with the original question (restricting the list of graphs based on "VertexConnectivity"), use Select:

Select[GraphData[], (GraphData[#, "VertexConnectivity"] >= 3) &]

However, this particular property apparently has some bugs, detailed at "Is there something wrong with VertexConnectivity". To be safe, we can use the function from Combinatorica, like so.

Block[{$ContextPath},
 Needs["Combinatorica`"];
 Needs["GraphUtilities`"]
 ]

Select[GraphData[],
 (Combinatorica`VertexConnectivity[
     GraphUtilities`ToCombinatoricaGraph[GraphData[#]]] >= 3) &]

This issue did not arise for me.

To combine multiple such conditions, use && between separate calls to GraphData, for instance:

Select[GraphData[], (GraphData[#, "EdgeChromaticNumber"] == 3 && GraphData[#, "VertexConnectivity"] >= 3) &]

All of these commands will take a very long time. It helps somewhat to preload all the information, getting all the downloads over with, which you can accomplish with GraphData[All, "Preload"]. But it still takes a long time to run. Restricting to graphs of a certain size makes it much quicker. To restrict to graphs on 7 vertices:

Select[GraphData[7], (GraphData[#, "EdgeChromaticNumber"] == 3 && GraphData[#, "VertexConnectivity"] >= 3) &]

You can also restrict this way based on the boolean properties, which GraphData can already do itself, as with "Nonplanar" in the question. Multiple ones may be combined like this:

Select[GraphData[{"Nonhamiltonian", "Nonplanar", "Connected"}, 7], (GraphData[#, "ChromaticNumber"] == 3) &]