  High-dimensional ($d\ge 5$) Ising systems have mean-field critical exponents.
However, at the critical temperature the finite-size scaling of the
susceptibility $\chi$ depends on the boundary conditions. A system with
periodic boundary conditions then has $\chi\propto L^{5/2}$. Deleting the
$5L^4$ boundary edges we receive a system with free boundary conditions and now
$\chi\propto L^2$. In the present work we find that deleting the $L^4$ boundary
edges along just one direction is enough to have the scaling $\chi\propto L^2$.
It also appears that deleting $L^3$ boundary edges results in an intermediate
scaling, here estimated to $\chi\propto L^{2.275}$. We also study how the
energy and magnetisation distributions change when deleting boundary edges.
