Eulerian and Hamiltonian Graphs · Challenge problem

The pentagonal prism graph (two $5$-cycles "top" $0,\ldots,4$ and "bottom" $5,\ldots,9$, joined by rungs $i\,(i+5)$) has a Hamiltonian cycle, as the top or bottom $5$-cycle alone shows. What is the minimum number of vertices that must be deleted from the full $10$-vertex graph so that NO Hamiltonian cycle survives among the remaining vertices and edges?