The crossed dodecahedral graph is the graph obtained from the dodecahedral graph by drawing edges between every pair of vertices in all its faces. It can therefore be constructed by adding edges in a pentagrammatic configuration to each face of a regular dodecahedron and is the dodecahedral analog of the 16-cell skeleton (which can be constructed by crossing the faces of a cube).
The graph is 2-planar, does not admit a straight-line 2-planar drawing, and has unique 2-planar embedding illustrated above (Brandenburg 2021, Bekos et al. 2017).
It will be implemented in a future version of the Wolfram Language as GraphData["CrossedDodecahedralGraph"].
