A crossed graph is the name given by Brandenburg (2021) to a graph obtained from a polyhedron by drawing edges between every pair of vertices in all its faces. This process leaves triangular faces unchanged.
The following table summarizes some special cases.
| base polyhedron | crossed graph | reference |
| cube | 16-cell graph | |
| cuboctahedron | octahedral line graph | |
| elongated square gyrobicupola | elongated square gyrorocupola graph | Fabrici and Madaras (2007) |
| regular dodecahedron | crossed dodecahedral graph | Brandenburg (2021) |
| icosidodecahedron | icosahedral line graph | |
| rhombic dodecahedron | crossed rhombic dodecahedron | |
| truncated tetrahedron | circulant graph  |
The crossed dodecahedral graph is the graph obtained by crossing the regular dodecahedron. It can therefore be constructed by adding edges in a pentagrammatic configuration to each face of a regular dodecahedron. It is also the graph square of the dodecahedral graph. This 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 is implemented in the Wolfram Language as GraphData["CrossedDodecahedralGraph"].
A different use of "crossed graph" appears in the notion of the crossed prism graph adopted in this work.
