The truncated dodecadodecahedral graph is the skeleton of the truncated dodecadodecahedron. It is illustrated above in a number of embeddings.
It is implemented in the Wolfram Language as GraphData["TruncatedDodecadodecahedralGraph"].
It appears to be difficult to find a unit-distance embedding for the truncated dodecadodecahedral graph due to vertices coinciding under iterative unit-distance refinement. For example, the drawings illustrated above all have unit edge lengths but also have overlapping vertices.
