Graphs from Chess Queens

On the GRAPHNET discussion board, Ed Pegg posed the following problem: find an arrangements of queens on a chessboard so that the queens correspond to vertices of the Petersen graph, with two queens adjacent if they are attacking each other. There are (at least) two ways to define the graph depending on whether or not we wish to allow queens to block the attack of other queens. Ed originally posed the blocking version, and presented a solution he and Daniele Giorgio Degiorgi found:
 Q Q _ Q _ _
 _ _ _ Q _ _
 Q _ _ _ _ Q
 _ _ _ Q _ _
 Q Q _ Q _ _
For the non blocking version, I found the following solution:
 Q Q _ _ _ _ _
 _ _ Q _ Q _ _
 _ _ _ _ _ _ _
 _ Q _ _ _ _ Q
 _ _ _ _ _ _ _
 Q _ _ Q _ _ _
 _ _ Q _ _ _ Q
Both of the above solutions are optimal in the sense that solutions on smaller boards do not exist. For the Heawood graph, I found the following solutions for the blocking and nonblocking variants of the problem.

Blocking:

 Q Q _ _ _ _ _
 _ _ _ _ _ Q _
 Q _ _ _ _ _ Q
 Q _ Q _ Q _ Q
 Q _ _ _ _ _ Q
 _ Q _ _ _ _ _
 _ _ _ _ _ Q Q
Nonblocking:
 Q Q _ _ _ _ _ _ _
 _ _ _ _ _ Q _ _ _
 _ Q _ _ _ _ Q _ _
 _ _ _ Q _ _ _ _ _
 Q _ _ _ _ _ Q _ _
 _ _ _ _ Q _ _ _ _
 _ _ _ Q _ _ _ _ Q
 _ _ _ _ _ Q _ _ Q
 _ _ _ _ Q _ _ _ _
The first of these is optimal in the sense that no solution on a smaller board exists. For the second, no solution on a smaller square board exists. A chess figure for the blocking solution is be found here.

Someone asked for a solution for the dodecahedral graph. Here's a blocking solution.

 Q Q Q Q _ _ Q _ Q _
 _ _ _ _ _ _ _ _ _ _
 _ _ _ _ _ _ _ Q _ _
 _ _ _ _ _ Q _ _ _ _
 Q _ _ _ _ _ _ _ _ Q
 _ _ _ _ _ Q _ _ _ _
 _ Q _ _ _ _ _ _ _ _
 Q _ _ _ _ _ _ _ _ _
 _ _ _ Q _ _ _ _ _ _
 Q _ _ Q _ _ Q Q Q Q

And here's one for the 4-cube:

 _ _ Q _ Q _ _ _
 _ _ Q _ Q _ _ _
 Q Q _ _ _ _ _ _
 _ _ _ _ _ _ Q Q
 Q Q _ _ _ _ _ _
 _ _ _ _ _ _ Q Q
 _ _ _ Q _ Q _ _
 _ _ _ Q _ Q _ _